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AN 

ELEMENTARY  TREATISE 

ON 

VARIABLE  QUANTITIES 


COPYRIGHT,  1921 

BY 
ASAHBL,  R.   COOK 


AN  ELEMENTARY  TREATISE  ON 

VARIABLE  QUANTITIES 

IN  TWO  PARTS: 

THE  DIRECT  AND  INVERSE 
By  HIRAM  COOK     V 


PRIVATELY  PRINTED 

LEDERER,  STREET  &  ZEUS  COMPANY 

BERKELEY.  CALIFORNIA 


Hiram  Cook  was  born  in  Preston,  New 
London  County,  Connecticut,  on  Decem- 
ber 11,  1827,  and  died  at  Norwich, 
Connecticut,  on  May  26,  1917.  This 
book,  published  after  his  death,  stands 
witness  to  his  lifelong  love  of  mathe- 
matics and  his  desire  to  put  his  knowledge 
in  this  field  at  the  service  of  others. 


16'649 


PREFACE 

The  subject  of  this  work  is  the  same  as  that  of  the 
Differential  and  Integral  Calculus,  but  parts  of  it  are  treated 
somewhat  differently,  especially  the  fundamental  principles; 
and  these,  it  is  believed,  are  made  so  clear  that  any  ordinary 
algebraic  student  can  readily  comprehend  them. 

In  regard  to  a  variable  quantity,  it  is  taken  to  mean  as 
qualified — that  is,  its  value  is  subject  to  a  continual  change, 
either  increasing  or  decreasing.  Now  this  being  the  case,  it 
is  evident  that  its  value  must  have  some  rate  of  increase  or 
decrease,  uniform  or  variable,  according  to  governing  condi- 
tions. Upon  this  theory  this  work  is  founded,  and  it  is  hoped 
it  so  clears  the  way  that  it  can  be  understandingly  followed 
by  those  who  are  so  inclined. 

How  is  it  in  regard  to  a  differential,  so  called,  and  the 
process  of  finding  it?  First  an  increment  is  added  to  the 
variable,  and  finally,  in  order  to  obtain  what  is  sought,  this 
increment  is  made  equal  to  zero  and  to  something  at  the  same 
time — the  something  being  taken  as  the  differential  of  the 
variable.  No  wonder  the  student  becomes  nonplussed,  for  it 
is  very  difficult  to  conceive  how  even  an  infinitesimal,  or  "the 
last  assignable  value  of  a  quantity"  and  zero  can  be  identical. 
Being  confronted  by  such  a  dilemma,  he  either  has  to  accept 
the  doctor's  diagnosis  or  give  the  matter  up  in  disgust. 

Let  it  not  be  imagined  that  this  work  is  claimed  to  be 
perfect  by  its  author,  or  that  he  considers  himself  more  than 
a  tyro  compared  with  the  great  mathematicians  of  the  past  or 
present.  He  simply  gives  his  theory  of  the  subject,  believing 
it  to  be  correct  and  both  reasonable  and  comprehensible,  and 
if  approved,  even  by  a  few,  he  will  not  feel  he  has  labored 
wholly  in  vain.  It  is  a  hard  matter,  however,  to  persuade  a 
man  to  part  with  his  idols ;  therefore,  since  Infinitesimal  was 
born  lang  syne  and  has  done  good  service,  possibly  it  is 
unreasonable  to  expect  that  the  little  fellow  should  be 
summarily  dismissed. 

Hiram  Cook. 

Norwich,  Connecticut 
1916 


CONTENTS  OF  PART  I 


Subject 


Article  Page 


Definitions 


ALGEBRAIC  FUNCTIONS 


Illustrations  of  rates    ...... 

Rate  of  u  =  a.t-"  .  .  .  . 

Rate  of  the  sum  or  difference  of  several  variables 

Rate  of  the  sum  or  difference  of  several  terms 

Rate  of  the  nth  power  of  tw^o  variables 

Rate  of  the  product  of  several  variables 

Rate  of  the  product  of  several  factors 

Rate  of  a  polynomial  ..... 

Rate  of  the  square  root  of  a  quantity 
Rate  of  a  fraction  .  .  . 

Successive  rates  and  ratal  coefficients     . 
Successive  rates  of  two  or  more  variables 
Special  rates  ...... 

Classified  rates  ...... 

Maclaurin's  and  Taj'lor's  Theorems 

LOGARITHMIC   FUNCTIONS 

Rate  of  u  =  a^'  and  x  =  log  tt      .         . 
Rate  oi  u=v->'      . 
Rates  relative  to  logarithmic  tables 
Illustration  of  principles  relative  to  curves 

CIRCULAR  FUNCTIONS 

Rates  of  sine,  cosine,  etc.,  in  terms  of  the  arc 
Rates  of  log  sine,  log  cosine,  etc.. 
Rate  of  the  arc  in  terms  of  sine,  cosine,  etc. 
Development  of  sin  x,  cos  x,  etc. 


1-8 


9 

S 

10 

6 

.  11 

8 

12 

8 

.  13 

9 

14 

9 

15-16 

10 

17 

11 

.  18 

11 

19 

12 

.  20 

13 

21 

13 

.  22 

15 

23 

15 

24-25 

16 

26-27 

20 

.  28 

22 

29 

23 

.  30 

25 

.  31 

26 

32 

29 

.  33 

30 

.    34 

32 

Value  of  11  =  R 


VANISHING  FRACTIONS 

{x  —  a)'"  x»  —  a»  tanji: 


(.1-  — a)' 


cot  2x 


-,  etc. 


35        33 


Signification  of  first  and  second  ratal  coefficients 
Curves  concave  and  convex  to  the  axis  of  x 
Ratal  equations  of  lines  of  the  first  and  second  order 
Tangents,  normals,  etc.,  of  curves        .... 

The  cycloid  ....... 

The  logarithmic  curve       ...... 

Asymptotes  ....... 

Rates  of  arc,  area,  etc.  ..... 

Radius  of  curvature  ...... 

Osculatory  circle  ...... 

Radius  of  curvature  of  lines  of  the  second  order 
Evolutes  and  involutes      ...... 

Curves  referred  to  polar  coordinates 

Tangents  and  normals  of  spirals  .... 

Rates  of  arc  and  area  of  spirals 

Radius  of  curvature  of  spirals  .... 

Singular  points  of  curves       ..... 

Maxima  and  minima  of  functions  of  a  single  variable 
Maxima  and  minima  of  functions  of  two  or  more  variables  67 


.  36 

38 

37 

39 

38-39 

41 

40-41 

43 

42-43 

46 

44 

47 

.   45 

49 

46-49 

52 

50-51 

54 

.  52 

57 

S3 

59 

54 

62 

55 

67 

56 

71 

.  57 

75 

58 

76 

59-63 

78 

64-66 

88 

95 


CONTENTS  OF  PART  II 


Subject  Article  Page 

Definitions  and  illustrations  .....    68-71      101 

SIMPLE  ALGEBRAIC  RATES 

Integral  of  a  monomial  rate      .....  72      102 

Integral  of  the  sum  or  difference  of  several  rates    .  .     73       103 

Integral  of  a  rate  of  the  form  du  =  (a  +  bx''^)^  x^-^dx        74      104 

Simple  circular  rates  ......      75-76      106 

Integration  by  series  ......  77      108 

BINOMIAL  RATES 

Any  binomial  rate  reduced  to  the  form 

du  =  (o  -f  bxn)r  x'^dx        78      111 

INTEGRATION   BY   PARTS 

Formulas  .4  and  5 79      113 

Formulas  C  and  D 80      118 

RATIONAL   FRACTIONAL   RATES 

When  the  factors  of  the  denominator  are  rational  .       81       119 

When  the  factors  of  the  denominator  are  imaginary      .        82      123 

Irrational  fractional  rates        ....  .        83-84      126 

TRANSCENDENTAL    RATES 

Exponential  rates  ...... 

Logarithmic  rates  ..... 

Circular  rates      ....... 

Bernouilli's  series      ...... 

Successive  integration  ..... 

Integration  of  partial  rates         .... 

Integration  of  total  rates      ..... 

Integration  of  homogeneous  rates 

Length  of  curves      ...... 

Area  of  curves  ...... 

Surface  of  revolution  ..... 

Volume  of  revolution  ..... 

Curved  surfaces  and  solids  referred  to  three  coordinate 
axes  ........ 

Curve  of  pursuit        ...... 


.  85 

130 

86 

132 

.  87 

134 

88 

136 

.  89 

138 

90 

140 

.  91 

141 

92 

143 

93 

144 

94 

149 

.  95 

153 

96-97 

157 

e 

.  98 

162 

99 

164 

PART  ONE 
DIRECT  METHOD 


PART  ONE 

DIRECT  METHOD 


DEFINITIONS 

Art.  1.  Two  classes  of  quantities  are  employed:  namely, 
constants  and  variables.  Constants  are  usually  represented  by 
the  first  letters  of  the  alphabet,  a,  b,  c,  etc.,  and  variables  by  the 
last,  u,  X,  y,  etc. 

The  value  of  a  constant  remains  the  same  throughout  the 
same  investigation ;  while  that  of  a  variable  continually  in- 
creases or  decreases  at  either  a  uniform  or  variable  rate. 

2.  The  variable  whose  rate  of  increase  or  decrease  is  as- 
sumed to  be  uniform  is  called  the  independent  variable,  and  the 
variable  whose  value  depends  on  that  of  the  independent  vari- 
able is  called  the  dependent  variable.  Thus  u  is  the  dependent 
and  X  the  independent  variable  in 

u  =  ax^  -{-  b. 

3.  The  dependent  variable  is  a  function  of  the  independent 
variable.    Thus  w  is  a  function  of  x  in 

u=^x^-\-  ax  -{-  b, 

which  is  expressed  generally  thus,  u  =  f(x),  in  which  f  is 
simply  a  symbol  denoting  function. 

4.  Functions  are  of  two  general  classes,  algebraic  and  trans- 
cendental. 

A  function  is  algebraic  when  the  dependent  variable  equals 
the  expression  containing  the  independent  variable  in  a  purely 
algebraic  form,  as 

u^a^  —  x^. 

A  function  is  transcendental  when  the  dependent  variable 
equals  the  expression  containing  the  independent  variable  in 


4  AN  ELEMENTARY  TREATISE 

the  form  of  an  exponent,  logarithm,  sine,  cosine,  tangent,  etc., 
as  in 

u  =  a^ ;  u^  log  x;  u  =  sin  x;  u  =  cos  x;  u  =  tan  x ,  etc. 

Transcendental  functions  are  of  two  classes,  logarithmic 
and  circular. 

5.  Functions  are  also  explicit,  implicit,  increasing,  and 
decreasing. 

An  explicit  function  is  one  in  which  the  dependent  variable 
is  directly  expressed  in  terms  of  the  independent  variable,  as  in 

u  =  ax^  -\-  b  OT  u  =  log  X. 

An  implicit  function  is  one  in  which  the  value  of  the  func- 
tion is  not  directly  expressed  in  terms  of  its  variable  and 
constants.    Thus  in  the  equation 

y^  -\-  axy  -\-bx^  -f-  c  =  0 

y  is  an  implicit  function  of  x — that  is,  y  is  not  directly  ex- 
pressed in  terms  of  x  and  the  constants  a,  b,  and  c. 

An  increasing  function  is  one  in  which  the  dependent 
variable  will  increase  when  the  independent  variable  increases, 
or  will  decrease  when  the  independent  variable  decreases,  as  in 

u  =  ax^  -\-  b. 

A  decreasing  function  is  one  in  which  the  dependent  vari- 
able will  increase  when  the  independent  variable  decreases,  or 
will  decrease  when  the  independent  variable  increases,  as  in 

1 

u^ — . 

X 

6.  A  function  may  consist  of  two  or  more  independent 
variables,  as 

w  =  ojir  ±  by  ±  C2  or  u  =  axy 2. 

7.  The  rate  of  a  variable — that  is,  its  rate  of  increase  or 
decrease — is  designated  by  writing  d  before  it,  as  du  represents 
the  rate  of  u,  dx  of  x,  dy  of  y,  etc. 

8.  A  ratal  coefficient  is  the  rate  of  the  dependent  variable 
.  du 

divided  by  that  of  the  independent  variable.     Thus  -—  is  the 

dx 

ratal  coefficient  of  w  =  /  (.*:). 


ON  VARIABLE  QUANTITIES 


Algebraic  Functions 
9.   Illustrations  of  the  application  of  the  rates  of  variables 


A 


B 


e: 


c 


o 


J 

F 


Fig.   1 

represent  the   area 
or  du,  the  rate  of  u. 


IS 


Let  the  side  AC  oi  the  rectangle 
ABCD  (Fig.  1)  be  represented  by  a, 
the  side  CD  by  x,  and  the  area  by 
u  =  ax.  Extend  AB  to  E,  CD  to  F, 
and  draw  EF  parallel  to  BD.  Now- 
let  dx,  the  rate  of  increase  of  x,  be 
represented  by  DF,  and  the  area  of 
ABCD  by  w  =  ax;  then  adx  will 

of  BEDF,  the  rate  of  increase  of  ABCD, 

Therefore  the  rate  of 

(1) 

(2) 


ax 


du  =  adx  =  ax^~'^  dx. 


Extend  AB  of  the  rectangle  ABCD  (Fig.  2)  to  i^  and  G, 
also  CD  to  L  and  H,  and  draw  KL,  EF,  and  G//  parallel  to 
AC.  Now  let  AC  he  represented  by  a,  FD  by  x,  CF  by  y,  and 
the  area  of  ABCD  by  ax  -j-  ay;  also  let  dx,  the  rate  of  ;ir,  be 
represented  by  DH,  and  dy,  the  rate  of  y,  by  LC ;  then  aa?;r  will 

represent  the  area  of  BGDH, 
the  rate  of  increase  of  the 
area  of  EBFD,  and  ady  will 
represent  the  area  of  KALC, 
the  rate  of  increase  of  the  area 
will  represent  the  rate  of  in- 
crease of  the  area  of  ABCD, 
of  AECF ;  hence  adx  -{-  ady 
Therefore  the  rate  of 

u=^ax  -{-  ay  (3) 

du  =  adx  -f-  ady.  (4) 

Extend  the  side  AB  of  the  rectangle  ABCD  (Fig.  3)  to 
G,  CD  to  H,  and  draw  EF,  KL,  and  GH  parallel  to  BD.  Now 
let  AC  ht  represented  by  a,  CD  by  x,  FD  by  y,  CF  by  jr  —  y, 
and  the  area  of  AECF  by  M  =  ajr — ay;  also  let  dx,  the  rate 

of  X,  be  represented  by  DH, 
and  afy,  the  rate  of  y,  by  Z)L; 
then  adx  will  represent  the 
area  of  BGDH,  the  rate  of  in- 
crease of  ABCD,  ady  that  of 
BKDL,  the  rate  of  increase 
of  EBFD,  and  adx— ady,  that 
of  KGLH,  the  rate  of  increase 
of  ^5CL)  less  that  of  EBFD, 


is 


Fig.  3 


6  AN  ELEMENTARY  TREATISE 

or  du,  the  rate  of  u.    Therefore  the  rate  of 

u  =  ax  —  ay  (5) 

is  du^adx  —  ady.  (6) 

Extend  the  side  AB  of  the  rectangle  ABCD  (Fig.  4)  to  G, 
CD  to  H,  AC  to  E,  ED  to  F,  and  draw  EE  parallel  to  AB, 
also  GH  to  i?D.    Now  let  CD  be  represented  by  ax,  AC  hy  y, 

and  the  area  of  ABCD 

by   li  =  a;r3r.      Let    adx, 

the  rate  of  a;?;,  be  repre- 

[^  iB         G  sented  by  DH,  and  (Z^', 


M 
E  F 


the  rate  of  y,   by  y4£; 
I        N  then  axdy  will  represent 

I  the  area  of  EEAB,  the 

I  rate  of   increase   of   the 

C  OH  area  of  ABCD  in  the  di- 

rection of  M,  and  aydx 
r  1  ^^    4-  ^jl[   represent   the   area 

of  BGDH,  the  rate  of  increase  of  the  area  of  ABCD  in  the 
direction  of  A''.  Hence  axdy  -f-  adyx  will  represent  the  total 
rate  of  increase  of  the  area  of  ABCD,  or  du,  the  rate  of  u. 
Therefore  the  rate  of 

u^axy  (7) 

is  du  =  axdy  -\-  aydx.  (8) 

10.  Let  y^x,  then  dy  =  dx.   Substituting  x  for  3;  in  (7)  of 
the  last  article,  also  x  for  3;  and  dx  for  dy  in  (8),  then 

w  =  ax^  ( 1 ) 

and  c?M  =  axdx  -\-  axdx  =  2axdx  =  2ax'^~'^dx.  (2) 

Let  y=^x'^,  then  (/y  ^  2jv(/jir.     Substituting  ;?;-  for  y  in  (7) 
of  the  last  article,  also  x^  for  y  and  2;ircf;ir  for  dy  in  (8),  then 

Mi^a^tr^  (3) 

and  du  =  2a.ar-flf;r  -\-  ax^dx  =  3ax^dx  =  3ax^~^dx.  (4) 

Let  y  =  :r^  then  dy^Sx^dx.     Substituting  as  before,  the 
rate  of 

u^:^ax^  (5) 

is  du  =  Ax^-^dx.  (6) 

Hence,  if  the  exponent  of  x  is  n,  n.  being  a  positive  integer, 

from  (1)  of  Art.  9  and  from  (2),  (4),  and  (6)  of  the  present 

article  it  is  evident  that  the  rate  of 

u  =  ax''  (7) 

is  du^:^anx^'~'^dx.  (8) 


ON  VARIABLE  QUANTITIES  7 

When  n  is  negative,  as  in 

u^ax-'^,  (9) 

multiplying  both  sides  by  jr"  gives 

ux'^  _—  ^ 

Passing  to  the  rate, 

x'^du  -\-  nux'^''^dx  =  0. 

Transposing  and  substituting  for  u  its  value, 

x^^du  =  —  anx~'^dx, 

and  dividing  by  x'\     du  =  —  anx~'^-^dx.  (10) 

When  the  exponent  of  ;ir  is  a  positive  fraction,  as  in 

u  =  ax''\  (11) 

raising  both  sides  of  the  equation  to  the  ^th  power, 

u?  =  a^x''. 

Passing  to  the  rate, 

su^~'^du  =  a^rx''~^dx.  (12) 

Raising  (11)  to  the  (s —  l)th  power  and  multiplying  by  s, 

j^s-i  ^  a'-^sx'-''/^,  (13) 

and  dividing  (12)  by  (13), 

r 
du^=a—x'''^-^dx.  (14) 

s 

When  the  exponent  of  ;ir  is  a  negative  fraction,  as  in 

u=^ax-'-'',  (15) 

raising  both  sides  of  the  equation  to  the  sih  power  and  multi- 
plying by  x'^  give 

u^x^'  =^  a*. 
Passing  to  the  rate, 

su'^''^x^du  -\-  ru^x'"'^dx  =  0. 
Transposing  and  dividing  by  su^-'^x''  give 

r 

du^  —  — ux'^dx, 
s 


or,  smce  u  =  ax~ 


r 
du^  —  a—x-'^/^-^dx.  (16) 

s 


8  AN  ELEMENTARY  TREATISE 

Hence,  the  rate  of  a  variable  affected  with  any  constant 
exponent,  -\-  or  — ,  having  also  any  constant  coefficient,  is  the 
product  of  the  coefficient  and  the  exponent  of  the  variable, 
multiplied  by  the  variable  with  its  exponent  less  unity,  into 
the  rate  of  the  variable. 

EXAMPLES 

1.  u  =  ax"-'^'^  2.  M  =  ax^''^ 

1 

3.    M^^7;ir(»+l)/n  4.    u  ^  —  CX-^ 

2 

11.  To  determine  the  rate  of  a  function  of  the  sum  or  dif- 
ference of  several  independent  variables,  as 

u  =  av  -\-  bx  -\-  cy  -}-  es,  ( 1 ) 

assume  u  =  r  -\-  s,  r  =  av  -\-  bx,  and  s  =  cy  -\-  ez,  the  rates  of 
which  [see  (4)  and  (6)  of  Art.  10]  are  respectively 

du  =  dr  -\-  ds,  dr  =  adv  +  bdx,  and  ds  ^=  cdy  -\-  edz.    (2) 

Substituting  the  values  of  dr  and  ds  in  du  ^  dr  -\-  ds  gives 

du  =  adv  -j-  bdx  -f-  cdy  -f-  edz.  (3) 

Hence  the  rate  of  the  sum  or  difference  of  several  inde- 
pendent variables  is  the  corresponding  sum  or  difference  of 
their  rates  taken  separately. 

12.  To  determine  the  rate  of 

u  =  ax  —  bx^  -f-  cx"^.  ( 1 ) 

Assume  v  =  ax,  y  =  bx',  and  z  =  ex"', 

the  rates  of  which  are  [see  (2)  of  Art.  9,  and  (2)  and  (10)  of 
Art.  10] 

dv  =  adx,  dy  =  2bxdx,  and  dz  =  cnx^^~'^dx.  (2) 

But,  according  to  the  assumption, 
M  =  t/  —  y  -\-  2, 

the  rate  of  which  is  [see  (1),  Art.  11] 

du^dv  —  dy-{-dz;  (3) 

therefore,  substituting  in  (3)  the  values  of  dv,  dy,  and  dz,  then 

du  =  adx  —  Zbxdx  -f-  cnx"-'^dx.  (4) 


ON  VARIABLE  QUANTITIES  9 

Hence  it  is  evident  that  the  rate  of  the  sum  or  difference  of 
any  number  of  terms  containing  the  same  independent  variable 
is  the  corresponding  sum  or  difference  of  their  rates  taken 
separately. 

13.  Required  the  rate  of 

u=  {ax  ±  byy.  (1) 

Assume  u  =  v^;  (2) 

then  v  =  ax±:by.  (3) 

Now  the  rate  of  (2),  from  Art.  10  is 

du  =  nV'-^dv,  (4) 

and  the  rate  of  (3)   [see  (3)  and  (5),  Art.  9]  is 

dv  ==  adx  dz  bdy. 

But  v'^~^  =  {ax  ±  by)'^-'^ ,  therefore,  by  substituting  in 
(4)  the  values  of  z^""^  and  dv,  the  result  is 

du  =  n  {ax  -\-  by)'^~'^  {adx  -\-  bdy)  (5) 

or  du  =  n  {ax  -f-  by)"'^adx  -\-  n  {ax  -\-  by)^^'^bdy.         (6) 

Hence,  the  rate  of  the  nth  power  of  the  sum  or  difference 
of  two  variables,  is  n  times  their  sum  or  difference  raised  to 
the  {n — \)th  power,  multiplied  by  the  sum,  or  difference  of 
their  rates,  whether  n  be  an  integer  or  fraction,  positive  or 
negative. 

EXAMPLES 

1 

I.  u  =  x^ —  —  x4-2x^^^  2.  u  =  x''-\-ax-'- ^b 

4 

3.  u  =  ax'^-'^  +  nx'^''^  4.  u=  {x  +  av)'*^^ 

14.  To  determine  the  rate  of 

u  =  vxy. 
Assume  s^xy;  ( 1 ) 

then  u  =  vs, 

and  the  rates  of  these,  from  Art.  9,  are 

ds  =  xdy  -\-  ydx  (2) 

and  du^vdz  -\-  zdv.  (3) 


10  AN  ELEMENTARY  TREATISE 

Substituting  the  value  of  z  from  (1);  and  ds  from  (2),  in 
(3),  then 

du  =  vxdy  -\-  vydx  -j-  xydv.  (4) 

Hence  it  is  evident  that  the  rate  of  the  product  of  any  num- 
ber of  variables  is  the  sum  of  the  products  obtained  by  multi- 
plying the  rate  of  each  variable  by  the  product  of  the  others. 

15.  To  determine  the  rate  of 

u  =  x''(a-\-x)  {bx^  -\-  cx^). 
Assume  v  =  a  -\-  x  (1) 

and  y==bx^-\-cx'^\  (2) 

then  u  =  x^vy. 

Passing  to  the  rate,  (1)  becomes 

dv  =  dx,  (3) 

(2),  by  Art.  12, 

dy  =  2bxdx  -f-  cnx^''^dx  =  (2bx  -{-  cnx"^-^)  dx,         (4) 
and  u  =  x'^'vy,  by  Art.  14, 

du  =  x^vdy  -\-  x^'ydv  -{-  rx^'^vydx.  ( 5 ) 

Substituting  the  values  of  v  and  y  from  (1)  and  (2),  also 
the  values  of  dv  and  dy  from  (3)  and  (4),  in  (5),  the  result  is 

du  =  X''  (a  -\-  x)  (2bx  -\-  cnx"-^)  dx  -\-  x''  (bx^  -f  cjr")  dx  + 

rx^~^  (a  -\-  x)  (bx'  -\-  ex")  dx, 

or  du^  {x''  {a-\-  x)  (2bx  -\-  cnx''-'^)  + 

X''  (bx^  +  ex'')  -j-  rx''-'^  (a^  x)  (bx^  +  ex"")  ]dx. 

Hence,  the  rate  of  the  product  of  any  number  of  factors 
containing  the  same  variable  is  the  sum  of  the  products  ob- 
tained by  multiplying  the  rate  of  each  factor  by  the  product  of 
the  others. 

16.  To  determine  the  rate  of 

u=:x^(a  —  z")  (b-\-y). 

Assume  v=a  —  s''^  ( 1 ) 

and  w^b-{-y^',  (2) 

then  M  =  xH'w.  ( 3  ) 


ON  VARIABLE  QUANTITIES  11 

The  rate  of  (1)  is 

dv  =  —  nz''-^dz,  (4) 

that  of  (2)  dw=^ry'-^dy,  (5) 

and  that  of  (3) 

du  =  x^vdw  +  x^wdv  -{-  2xvwdx.  (6) 

Substituting  the  values  of  v  and  w  from  (1)  and  (2),  also 
the  values  of  dv  and  dw  from  (4)  and  (5),  in  (6)  gives 
du  =  rx^  (a  —  2"")  y-^dy  —  nx^  (b  -{-  y )  z'^-'^dz  -\- 
2x  (&  +  3''")  (a  —  ^")  dx. 

Hence  the  preceding  rule  is  also  applicable  when  each  fac- 
tor contains  a  different  variable,  or,  as  is  evident,  even  when 
each  factor  contains  several  variables. 

EXAMPLES 

1.  u^x^yz'^  2.  u=  {bx  -\-  c)  {x" -\- ax) 

3.  u  =  x^  {y^  -\-  av) 

17.  To  determine  the  rate  of 

u=  {a  -{-  bx  -{-  ex- ) ". 
Assume  y  =  a  -\-  bx  -\-  cx^ ;  ( 1 ) 

then  u  ==  y^. 

Passing  to  the  rate, 

dy^^{b-\-2cx)dx  (2) 

and  du  =  ny^''^dy.  (3) 

Substituting  the  value  of  y  from  (1),  and  dy  from  (2),  in 
(3),  then 

du  =  n  (a  -\-  bx  -{-  cx^)"''^  (^  +  2cx)  dx. 

Hence,  the  rate  of  a  polynomial  affected  with  atiy  constant 
exponent  is  the  exponent  into  the  polynomial  with  its  ex- 
ponent less  unity,  multiplied  by  the  rate  of  the  polynomial. 

18.  To  determine  the  rate  of 

u^\/ {ax  -\-  bx^) 
or  M  ^  (ax  +  bx^)^''^. 


12  AN  ELEMENTARY  TREATISE 

Passing  to  the  rate,  as  in  Art.  17, 

1 

du  =  —  {ax  -\-  bx'^)~'^  (o  +  nbx'^~^)  dx, 

(a  -}-  nbx"~^)  dx        (a  -{-  nbx^^~^)  dx 
or  du  = ^= . 

2  {ax  +  bx" y^  2  V («■*■  +  bx" ) 

Hence,  the  rate  of  the  square  root  of  a  quantity  is  the  rate 
of  the  quantity  under  the  radical,  divided  by  twice  the  radical. 

19.  To  determine  the  rate  of  a  fraction,  as  the  function 

V 

z 
Multiplying  through  hy  z,  uz=^v; 

then  passing  to  the  rate,  by  (7)  of  Art.  9, 
udz  -\-  zdu  ==  dv. 
Substituting  for  u  its  value  and  transposing  give 

vdz 

zdu  f^dv  — 

z 

zdv  —  vdz 


or  zdu ■ 


z 
Therefore,  dividing  by  z, 


zdv  —  vdz 
du=^ '. 


Hence  the  rate  of  a  fraction  is  the  denominator  into  the 
rate  of  the  numerator,  minus  the  numerator  into  the  rate  of 
the  denominator,  divided  by  the  square  of  the  denominator. 

If  2/  be  a  constant,  then,  since  a  constant  has  no  rate, 

vdz 

du^^^  — ; 

z'~ 

that  is,  when  t/  is  a  constant,  w  is  a  decreasing  function  of  z 
and  its  rate  is  consequently  negative. 

EXAMPLES 

1.  u^^{l  -\-  x^)  3.  M=;tr«  (.^  —  a)  (a  —  x-) 


2.  u  =  ^y(x^-\-y^)         4.  u 


VGr+1)— V(.r— 1) 


ON  VARIABLE  QUANTITIES  12 

Successive  Rates  and  Ratal  Coefficients 
20.  In  obtaining  these,  and  at  the  same  time  to  exemplify 
the  work,  let 

u^x''  -\-  ax^.  ( 1 ) 

Passing  to  the  rate,  by  Art.  12, 

du=  (nx^^~'^  -j-  2ax)  dx.  (2) 

Passing  to  the  rate  again,  regarding  dx  as  constant, 

d  (du)  =d^u={n  (n—1)  x""-^  ^2a]dx\  (3) 

In  like  manner  it  will  be  found  from  (3)  that 

d^u  =  n  (n  —  1 )  (n  —  2)  x'^'^dx^.  (4) 

(2),    (3),    and    (4)    are    successive    rates    of    (1),    and 
(^^n-i  _^  2ax),  [n  (n —  1)  x"-'^  +  2a} 

and  {n  (n — 1)  (n  —  2)  .r"~^}  are  respectively  coefficients  of 
dx,  dx^,  and  dx^. 

Dividing  (2)  by  dx,  (3)  by  dx'^,  and  (4)  by  dx^,  the  results 


are 


du 


=  n.r""^  -\-  2ax 

dx 

(5) 

d^u 

=  n  (n  —  1 )  x"-^  +  2a 

dx'- 

(6) 

d'u 

=  n(n — 1)  (n  —  2)x''-\ 

dx' 

(7) 

and 


du     d^u  d'u 

Inasmuch  as  , ,  and  are  respectively  equal  to 

dx    dx^  dx' 

the  coefficients  of  dx,  dx^,  and  dx',  they  are  called  ratal  co- 
efficients ;  du,  d^u,  and  d'u  are  the  first,  second,  and  third  rates 
of  the  dependent  variable  u,  and  dx,  dx^,  and  dx'  are  the  first, 
second,  and  third  powers  of  the  rate  of  the  independent  vari- 
able X. 

Rates  of  Functions 
OF  Two  OR  More  Independent  Variables 

21.  It  has  been  shown  in  Art.  9  that  the  rate  of  u  =  xy  is 

du  =  ydx  -\-  xdy ;  therefore, 


14  AN  ELEMENTARY  TREATISE 

if  M  =  x'^y, 

its  rate  is  du  =  nx'^-'^ydx  +  ji;"c?3; ; 

and  if  u  =  x''y'^, 

its  rate  is  a?w  ^  M;ir""^3;'^c?;ir  +  mx^y'^'''^dy  ; 

also  if  M  ^  ;r«3;'«  +  ;ir'"3'^                                    (1) 

its  rate  is 

du  =  nx^'-'^y'^dx  +  Wjir'^3;"»-iflf3;  +  rx'-^y'dx  +  j;»r''y«-^fl?3'.  (2) 

See  preceding  rules. 

Now  if  the  rates  of  ( 1 )  be  first  taken  under  the  supposition 
that  X  varies  and  y  remains  constant,  then  that  y  varies  and  x 
remains  constant,  the  sum  of  the  results  will  be  the  same  as 
(2) :  thus 

du  =  nx'^'^y^dx  -\-  rx^'^y^dx  (3) 

and  du  =  mx^'y^"'~'^dy  -{-  sx^'y^'^dy.  (4) 

Adding  (3)  and  (4), 

du  =  nx"'^y"^dx  +  rx'"^y^dx  -\-  mx'^y^'^dy  -f-  sx''y^~^dy.  (2) 

Dividing  (3)  hy  dx  and  (4)  by  dy,  the  results  are 

du 

=  nx"-'^y"'  +  rx''-^y^  ( 5  ) 

dx 

du 

and  =  mx'^y^"'''^  -\-  sx''y^~'^.  (6) 

dy 

By  taking  the  rate  of  (5)  with  respect  to  y  and  the  rate  of 
(6)  with  respect  to  x,  the  following  are  found, 

d^u 

=  mnx"-'^y"'-^  +  rsx^'-'^y^-^  (/) 

dxdy 

d^u 

and  =  mnx^'-'^y'"-'^  +  rsx''-'^y^-^  (8) 

dydx 

in  which  the  right-hand  members  are  identical ;  therefore 

d^u  d'^u 

dxdy      dydx 

(2)  is  called  the  total  rate,  (3)  and  (4)  partial  rates,  and 
(5)  and  (6)  ratal  coefficients.  The  second,  third,  and  higher 
rates  can  be  found  in  a  similar  manner  as  in  Art.  20. 


ON  VARIABLE  QUANTITIES  15 

If  the  function  contains  three  independent  variables,  as 

by  proceeding  in  like  manner  the   following  results  will  be 
obtained : 

d^u  d^u         d^u  d^u  d'U  d^u 

dxdy       dydx      dxdz       dzdx      dydz        dzdy 

If  the  function  contains  four  independent  variables,  there 
will  be  six  of  these  equalities,  if  five  there  will  be  ten,  and  so  on. 

Special  Rates 

22.  Let  u=  {x  ^yY,  (1) 

then,  by  taking  the  rate  first  with  respect  to  x  and  secondly 
with  respect  to  y,  the  following  are  found, 

du^n{x  -^yY'^dx  (2) 

and  du^=n  {x -\- yy^^'^dy.  (3) 

Dividing  the  first  by  dx  and  the  second  by  dy  gives 

du 

— -  =  w(^  +  y)"-i  (4) 

dx 

du 

and  ——^=n{x-\-yy-'^  (5) 

dy 

in  which  it  will  be  observed  that  the  right-hand  members  are 
identical. 

The  sum  of  (2)  and  (3)  is 

du=^n  (x  -\-  y)^~'^(dx  +  dy), 

virtually  the  same  as  given  in  Art.  13. 

Classified  Rates 

23.  When  the  partial  rates  of  a  function  of  two  or  more 
independent  variables  are  taken  with  respect  to  one  variable 
only,  they  are  said  to  be  of  the  first  class;  when  taken  with 
respect  to  one  variable  and  that  rate  taken  with  respect  to  an- 
other variable,  they  are  said  to  be  of  the  second  class :  thus,  if 

u  =  x^y^  +  x^y,  ( 1 ) 

by  taking  the  rate  with  respect  to  x  only,  the  results  are 

du  =  Sx^y^dx  -\-  2xydx 

d^u  =  6xy^dx^  -\-  2ydx^ 
and  d^u  =  Gy^dx^, 

which  are  partial  rates  of  the  first  class. 


16  AN  ELEMENTARY  TREATISE 

Taking  the  rate  of  (1)  with  respect  to  ;r  gives 

du  =  Sx^y'^dx  +  2xydx  (2) 

and  this  rate  taken  with  respect  to  y  is 

d^u  ==  6x^ydxdy  -\-  2xdxdy ;  (3) 

(2)  and  (3)  are  partial  rates  of  the  second  class, 

Maclaurin's  Theorem 
This  theorem  explains  the  method  of  developing  into  a 
series  a  function  of  a  single  independent  variable. 

24.  Assume  the  development  to  be 

f  {x):=A-^Bx-^Cx^~-{-Dx^-^ttc.,  (1) 

in  which  A,  B,  C,  D,  etc.  are  constants  whose  values  depend 
entirely  upon  those  which  enter  /  {x). 

Now  in  order  to  determine  the  values  of  A,  B,  C,  D,  etc. 
such  as  will  render  the  assumed  development  true  for  all 
values  of  x,  let 

u  =  A-\-Bx^Cx^"^Dx^^e\.c.  (2) 

and  of  this  find  the  ratal  coefificient,  as  in  Art.  20;  thus 

du 

=  5  +  2Cjr  +  3Z).s;- +  etc.  (3) 

dx 

d^u 


dx^ 
d^u 


2C  +  2  •  3Dx  +  etc.  (4) 

2  ■  3D -\- etc.  (5) 

dx' 

Making  .ar  =  0,  it  will  be  found  from  (3),  (4),  (5),  etc.  that 

du  1     d^u  1       d'u 

B=—,     C  = ,      \D  = ,etc.  (6) 

dx  2     dx^  2-3    dx' 

Since  A  will  retain  the  same  value,  whatever  the  value  of  x, 
substituting  the  values  of  B,  C,  D,  etc.  in  (2)  will  give 

xdu  x^d^u  x'd'u 

u  =  A^ + + +  etc.,  (7) 

dx         1  ■  2dx^       1  •  2  •  3dx' 

the  theorem  of  Maclaurin. 

If  the  exponent  of  the  variable  is  greater  than  unity,  as 
M=  (a  -)-  bx^),  assume  bx^^v;  then  substitute  for  v  and  its 
rates  their  values  in  the  development  of  m  =(a  -|-  t/). 


ON  VARIABLE  QUANTITIES  17 

When  the  function  or  any  of  its  ratal  coefficients  becomes 
infinite  by  making  its  variable  equal  to  zero,  it  can  not  be 
developed  by  this  theorem — as,  for  instance,  u  =  ax^''. 

For  an  exemplification  of  this  theorem,  take 

u^={a-^  xY.  '       (1) 

Determining  the  ratal  coefficient,  as  in  Art.  20, 

=  M(a  +  ;ir)«-i  (2) 


dx 


n  {n—l)  {a^xy-^  (3) 


d^u 


dx- 

n  {n — 1)  (w  —  2)  (a +  ;^) ''-3,  etc.  (4) 


dx 

Making  x  =  0,  then  from  (2),  (3),  and  (4)  are  found 
du 
dx 

d^u 

=  n  (n — 1)  a^-^ 

dx'  ^ 

d^u 

n{n  —  1)  (n  —  2)  a""^ 


dx^ 

and  from  ( 1 ) ,  when  jr  =  0,  w  =  a"  :   that  is,  A  =  a". 

Substituting  these  values  in  (7)  will  give 

na'^~'^x         n  (n —  1)  aP~'x' 

+ ^-^ + 


1  1-2 

n  {n  —  1)  (w  —  2)  a'^'^x 


+  etc., 


1-2-3 
the  same  as  found  by  the  binomial  theorem, 

EXAMPLES 

\.u={l-\-xY  2.u={a^bx)-' 

Taylor's  Theorem 
25.  This  theorem  explains  the  method  of  developing  into 
a  series  any  function  of  the  sum  or  difference  of  two  inde- 
pendent variables,  according  to  the  ascending  powers  of  one 
of  them. 


18  AN  ELEMENTARY  TREATISE 

Let  u  =  f{x  +  y)  (1) 

and  assume  the  development  to  be 

u  =  A^By-^Cy'  +  Dy^-^  etc.,  (2) 

in  which  A,  B,  C,  D,  etc.  are  functions  of  x. 

Now  in  order  to  find  the  values  of  A,  B,  C,  D,  etc.,  such  as 
will  render  the  development  true  for  all  possible  values  which 
may  be  ascribed  to  x  and  y,  determine  the  ratal  coefficients  of 
(1),  first  under  the  supposition  that  x  varies  and  y  remains 
constant,  then  that  y  varies  and  x  remains  constant.  By  this 
process  it  will  be  found  that 

du        dA         dB  dC  dD 

=^ + y  -h /  + /  +  etc.     (3) 

dx        dx        dx  dx  dx 

and        -^  =  B  ^  2Cy  +  Wy'  +  etc.,         (4) 

dy 

but  since  these  ratal  coefficients  are  identical   [see    (4)    and 
(5)  of  Art.  22]  it  follows  that 

dA        dB  dC  dD 

B^ZCy-^  3Df-  =  --  +  -—y  +  —-y'  +  -—y'+  etc.,  (5) 
dx  dx  dx  dx 

in  which  the  coefficients  of  like  powers  of  3;  must  evidently  be 
equal — that  is, 

dA  dB  dC 

B  = ,  C  =  —  and  £>  = ,  (6) 

dx  2dx  3dx 

the  rates  of  which  are,  regarding  dx  as  constant, 

d-A                 d~B                 dK 
dB= ,     dC  = ,     dD  = ;  (7) 

dx  2dx  3dx 

d^A                  d'B 
also  d^B  = ,    d'C  = .  (7) 

dx  2dx 

From  (6)  and  (7)  it  will  be  readily  found  that 

dA  d-A  d^B  d^A 


B= .     C  = and  D 


dx  2dx-  2-3dx^       2-3dx^ 

Substituting  these  values  of  B,  C,  and  D  in  (2)  will  give 

dA  d^A  d^A 

u  =  A-\- 3;  + y-  + /  +  etc.,  (8) 

dx        2dx^  2  ■  3dx^ 

known  as  the  theorem  of  Taylor. 


ON  VARIABLE  QUANTITIES  19 

In  like  manner  the  development  of  u=^f  {x  —  y)  will  be 
found  to  be 

dA  d'A  d'A     ^   , 

u  =  A— 3;  + /  —  — :; y  +  etc.         (9) 

dx  2dx^  2  ■  3dx^ 

Although  this  theorem  gives  the  general  development  of 
every  function  of  the  sum  or  difference  of  two  variables  cor- 
rectly, yet  in  some  particular  cases  a  certain  value  may  be 
ascribed  to  the  variable  x  which  will  render  the  development 
impossible,  as  will  be  indicated  by  some  of  the  ratal  coefficients 
of   the   development   becoming  equal   to   infinity :   thus,   if   in 

u^a  -\-  (b  -\-  X  —  y)^''^ 

y  be  made  equal  to  zero,  then 

A  =  a^  (b-\-xy^, 

the  first  and  second  ratal  coefficients  of  which  are 

dA  1  d^A  1 

and 


dx         2{b-\-xy^  dx-  4r{b-^xY'^ 

both  of  which  become  equal  to  infinity  when  x=^  —  b. 

For  an  exemplification  of  the  theorem,  develop 

u=  {x  -\-  yy\ 
Making  3;  =  0  gives 

u^A=^  x'\ 

the  successive  ratal  coefficients  of  which  are,  from  Art.  20, 

dA  d^A 

nx^~'^,     =  M  (m — 1)  x"'~, 


dx  dx 

d^A 


^^n  (n — 1)  (n  —  2)  x"'^,  etc. 
dx' 

Substituting  these  values  in  formula  (8),  the  result  is 

n  (n —  1) 
u  =  x''  -\-  nx'^-^y  -\- x"-^y-  -f 

n  (n —  1)  (w  —  2) 

x"-"y'  -\-  etc.. 


2-3 
the  same  as  found  by  the  binomial  theorem, 

EXAMPLES 

I.  u=  (x -\- yy^  2.  u  =:^  (x -\- ay)- 


20  AN  ELEMENTARY  TREATISE 

Transcendental  Functions 
26.  Let  the  function  be 

u  =  a^.  (1) 

Assuming  a  =  \  -\-  c, 

then  M^  (1  -f  c)^, 

the  development  of  which,  by  the  binomial  theorem,  is 

X  {x — 1)  X  {x — 1)  {x  —  2) 

u^l  -\-  xc  -\- c~  -\- c^  4- 

2  2-3 

x{x—l)  {x  —  2)  {x  —  Z) 

c^  +  etc.  (2) 


2-3-4 
Passing  to  the  rate, 

2x  —  1  Zx^  —  6;r  -|-  2 

2  2-3 

Ax^  —  l^x^  +  22x  —  6 

c^  +  etc.)  dx.  (3) 

2-3-4 

Dividing  each  member  of  (3)  by  the  corresponding  mem- 
bers of  (2)  gives 

du  c^       c^        c^ 

=  (c  —  — +  —  —  — -|-etc.)  a?;ir,  (4) 

w  2        3         4 

which  is  the  ordinary  logarithmic  series  for  log  (1-|-  c),  that 

c^       c"        c* 
is  log  (1 +  c)  =c  — — +  — —  — +  etc.  (5) 

2       3        4 

But  \  Ar  c^=-a,  therefore 

du 

=  dx  log  a. 

u 

Substituting  for  u  its  value  and  multiplying  by  a^, 

du^a^dx  log  a.  (6) 

Substituting  in  (4)  for  u  and  c  their  values  and  multiplying 
by  a^,  the  result  is 

a—\         {a  —  \y 

du  =  a-''  ( — -\- 

1  2 

(a._l)3        {a—iy 

— +  etc. )  dx. 

3  4 


ON  VARIABLE  QUANTITIES  21 

Assuming 

a— I         {a—\y        {a—\y       {a—iy 

— -\-  etc.  =^ 


12  3 

then  du^a'^  edx,  (J) 

wherein  e  is  dependent  on  a  for  its  value,  which  may  be  de- 
termined by  Maclaurin's  theorem. 

Since  the  rate  of  u^a'^  is  du=^a''edx,  (7),  it  is  evident, 
e  being  constant,  that  the  rate  of  du  =  a'^edx  is  d^u  =^  a^e'^dx^ 
and  the  rate  of  d^u  ^=  a-'e^dx^  is  d^u  =  a'^e^dx^.  Therefore, 
the  ratal  coefficients  are,  from  Art.  20, 

du  d'^u  d'^U' 

dx  dx^  dx^ 

Making  x^O  in  (1),  also  in  the  ratal  coefficients,  it  will 
be  found  that 

du  d^u  d^u 

u=^l,     =  ^,     ^e'^,     ^^^,  etc. 

dx  dx~  dx^ 

Substituting  these  values  in  (7),  Art.  24,  gives 

ex  e'^x'^  e^x^ 

li  =  a^  =  1  4- -\- -|- +  etc. 

1  2  2-3 

1 

If  .ar  =  — ,  then 
e 

1         1  1 

a*-  ==  1  _u  —  ^  —  -)- -1-  etc. 

12         2-3 

The  sum  of  the  first  twelve  terms  of  this  series  is  2.7182818, 
which  is  the  base  of  the  Naperian  system  of  logarithms ;  hence 
e  is  the  Naperian  logarithm  of  a.  Therefore,  substituting 
log  a  for  e  in  (7),  then 

du  =  a^dx  log  a, 
the  same  as  (6). 

Hence  the  rate  of  an  exponential  function  is  the  function 
into  the  rate  of  the  exponent  multiplied  by  the  Naperian  log- 
arithm of  the  constant  of  which  the  variable  is  the  exponent. 


22  AN  ELEMENTARY  TREATISE 

27.  Resuming  (1)  of  the  last  article,  transposing  it,  and 
taking  the  logarithm  of  both  members  give 

ji:  log  a  :=  log  u, 

log  u 

whence  x  = .  ( 1 ) 

log  a 

Now  it  has  been  shown  in  the  last  article,  that  the  rate  of 

M  =  a*  is 

du  =  a^dx  log  a, 
du 

whence  dx  = 

a'^  log  a 

or,  substituting  for  a^  its  value,  from  (1)  of  Art.  26, 

du 

dx  = ,  (2) 

u  log  a 
the  rate  of  (1). 

If  a  is  the  base  of  a  system  of  logarithms,  then  x  is  the 

1 

logarithm  of  u  in  that  system,  and is  the  modulus  of  the 

log  a 

1 

system;  therefore  representing by  M,  (2)  becomes 

log  a 

du 
dx  =  M .  (3) 

u 

Hence  the  rate  of  the  logarithm  of  a  quantity  is  the  modulus 
of  the  system  into  the  rate  of  the  quantity,  divided  by  the 
quantity  itself. 

The  modulus  of  the  Naperian  system  of  logarithms  is 
unity;  therefore  if  the  logarithms  are  taken  in  the  Naperian 
system  (3)  becomes 

du 
dx  = . 

u 

Hence  the  rate  of  the  Naperian  logarithm  of  a  quantity  is 
the  rate  of  the  quantity  divided  by  the  quantity  itself. 

28.  To  determine  the  rate  of 

u^v^, 
in  which  both  v  and  x  are  variables. 


ON  VARIABLE  QUANTITIES  23 

Taking  the  logarithm  of  both  members, 
log  M  =  ^  log  V, 
and  passing  to  the  rate,  by  Arts.  26  and  9, 
du       xdv 


-j-  dx  log  V, 
u  V 

uxdv 
or  du  = -|-  udx  log  v. 

V 

Substituting  t"^  for  u  and  reducing, 

du  =  xv^~'^dv  -f  v^dx  log  v.  ( 1 ) 

Hence  the  rate  of  a  variable  quantity  having  a  variable 
exponent  is  the  sum  of  the  rates  obtained,  first  under  the  sup- 
position that  the  quantity  varies  and  the  exponent  remains 
constant,  then  that  the  exponent  varies  and  the  quantity  remains 
constant. 

29.    Take  u  =  \og{\^x),  (1) 

in  which  u  is  the  Naperian  logarithm  of  1  +  -^^  ^^^  the  suc- 
cessive ratal  coefficients  are 

du  1  d'^u  1  d^u  1  •  2 

-,  etc. 


dx         \-^x     dx^  (1+^)'     dx^        (1+-^)' 

Making  ;r  ^  0  in  ( 1 ) ,  also  in  the  ratal  coefficients,  then 

du  d^u 

dx  dx~ 

d^u  d*u 

=1-2,     =  —1  •2-3,  etc. 

dx"  dx^ 

and  consequently,  by  substitution  in  (7)  of  Art.  24, 

x'^       x"       X* 
u^los:  (I  4- x)  =x  —  — -|- —  —  —  +  etc.  (2) 

2        3       4 

[see  (5)  of  Art.  26]. 

Developing  u  =  log  (1  —  x) 

in  like  manner,  it  will  be  found  that 

X^         X^'         X* 

M^logfl — x)  <=  —  X  —  —  —  —  —  —  —  etc.      (3) 

^  V  ;  2        3        4  ^ 


1  +  ^ 

v-\-  1 

1—x 

V    *  ' 

1 

24  AN  ELEMENTARY  TREATISE 

Subtracting  (3)  from  (2)  gives 

log(l+^)— log(l— ;r)=2(^  +  ^  +  y  +  etc.)(4) 

1  A-x         v+  1 
Assuming 

then 

2v-\-l' 

therefore,  since 

log  ( )  =  log  (z/  +  1 )  —  log  V, 

V 

by  substituting  the  value  of  x  in  (4),  the  following  is  obtained : 

log  {v^  1)— logz/  = 

1  1  1 

2  [ _|- 4- ^  etc.], 

(2z/+l)         2>{2v^iy        5(2z/+l)s 

or  log  (z/ +  1)  ^log^ -f 

1  1  1 

2  [ + + +  etc.], 

(2z/+l)         3(2z/+l)3         5(2z/-fl)5 

by  which  the  logarithm  of  ^'  -|-  1  can  readily  be  found  when 
the  logarithm  of  v  is  known.     Thus,  if 

111 
v=\,  log 2=       0  +  2(— + + +  etc. )=  0.69314718 

^  3       3  •  33     5  •  3^ 

1  1  1 

v=2,  log  3  =  log  2  +  2(— + + +  etc.)=  1.09861229 

t;=3,  log  4  =  log  2  +  log  2  =  1.38629436 

I  1  1 

v=^,  log  5  =  log  4  +  2(— + + +  etc.)=  1.60943791 

^  ^  ^93- 93     5- 9^ 

v=S,  log  6  =  log  2  +  log  3  =  1.79175947 

II  1 

v=6,  log  7  =  log  6  4-  2(— + + +  etc.)=  1.94591014 

^  ^     ^    ^3    3-13^     5-13^ 

t/=7,  log  8  =  log  2  + log  4  =2.07944154 

t;=8,  log  9  =  log  3  +  log  3  =  2.19722458 

v=9,  log  10=-log  2  +  log  5  =  2.30258509 


ON  VARIABLE  QUANTITIES  25 

The  logarithm  of  10  in  the  common  system  is  1,  and  in  the 
Naperian  system  is  2.30258509;  1  divided  by  2.30258509  is 
0.43429448,  the  modulus  of  the  common  system,  usually  desig- 
nated by  M. 

To  avoid  inconvenience,  the  Naperian  logarithms  are  gen- 
erally used  in  this  work.  Whenever  the  common  system  may 
desired,  it  will  be  necessary  to  multiply  by  the  modulus  of  that 
system. 

EXAMPLES 

To  determine  the  rate  of 

X 

u  =  log . 

X 

Assuming  u  = ,  ( 1 ) 

(a^  +  x^y^ 

dv 
then  u  =  \ogv  and  du  = .  (2) 

V 

From  ( 1 ) ,  by  passing  to  the  rate, 

(a^ -\- x^y  dx  —  x^  {a" -\- x^)-^- dx 


dv 


{a^  -f  x^) 


a^dx 
or,  reducing,  dv  =  — — — -— .  (3) 

{or  -\-  x^Y'^ 

Substituting  for  v  and  dv  their  values  in  (2)  gives 

a^dx 


du  = 


X  (a^  -f-  x^) 


a^  +  b''  log  X 

2. 


a'^  -\-  b^  log  y 

(l+;,)    +    (l+3,) 

6.  u  =  log 

(1+.^)  — (l+y) 

What  is  the  rate  of  the  common  logarithm  4300? 

Illustrations  of  Principles  Relative  to  Curves 
30.    If  a  particle  impelled  from  A  toward  B  along  the  curve 


26 


AN  ELEMENTARY  TREATISE 


Y 


A 


E 

f; 

^' 

P 

G 

'B 


D 
Fi  a.  3 


APB   (see  figure)  be  left  to  itself  at  any  point  in  the  curve, 

Q  as    at   P,    it   is    obvious    that 

it  would  then  proceed  at  a 
uniform  rate  toward  C  along 
the  straight  line  PC  tangent 
to  the  curve  at  P. 

For  the  point  P,  let  x  re- 
present the  abscissa  AD,  y  the 
ordinate  DP,  and  z  the  curve 
AP.     Extend  DP  to  E,  and 
jx:^  draw  EF  and  PG  parallel  to 

AX]  also  draw  £F  and  EG 
parallel  to  AY.    Then  dx  will 
be  represented  by  PG,  dy  by 
^P,  and  ds  by  PF ;  hence 
dz'  =  dx^  -\-  dy~  or  dz  =  ((/;r-  -[-  dy~Y^. 

Circular  Functions 
To  determine  the  rate  of 

u  ;=:  sin  X, 
let    the    radius    AC  =BC  =  R    (see 
figure),  the  arc  AB  =  x,  and,  as  the 
case  may  be,  let  u  represent  the  sine, 
cosine,    tangent,    etc.    of    the    arc    x; 
then  we  have,  by  Art.  30,  BE  =  dx, 
BD  equal  to  the  sine,  CD  the  cosine, 
EF  the  rate  of  the  sine,  and  BF  the 
rate  of  the  cosine ;    hence 
BC:CD::BE:EF 
or  R  :  cos  x::dx:  du, 

cos  xdx 

du^ . 

R 


31. 


O  A 


Fi  a.   6 


whence 


Therefore  the  rate  of  the  sine  of  an  arc  is  equal  to  the 
cosine  of  the  arc  into  the  rate  of  the  arc,  divided  by  the  radius. 

If  u^  cos  X, 

then,  since  u,  the  cosine  CD,  is  a  decreasing  function  of  the 
arc  X,  its  rate  is  negative ;  therefore 

sin  xdx 


R:  sin  xwdx:  —  du  or  du 


R 


ON  VARIABLE  QUANTITIES  27 

Hence  the  rate  of  the  cosine  is  minus  the  sine  into  the  rate 
of  the  arc,  divided  by  the  radius. 

If  tt  =  vers  x^^R  —  cos  x, 

sin  xdx 

then  du  =^ . 

R 

Hence  the  rate  of  the  versed  sine  is  the  sine  into  the  rate  of 
the  arc,  divided  by  the  radius. 

R  sin  X 
If  w  =  tan^^ , 

cos  X 

then,  by  passing  to  the  rate,  by  Art.  19, 

COS"  xdx  -\-  sin-  xdx 

du^ , 

cos^  X 

or,  since  cos-  x  -\-  sin-  x  =  R'~, 

R^dx 


du 


cos-  X 

Therefore  the  rate  of  the  tangent  of  an  arc  is  equal  to  the 
square  of  the  radius  into  the  rate  of  the  arc,  divided  by  the 
square  of  the  cosine. 

R  cos  X 

If  w  =  cot  X  = , 

sin;ir 

then,  passing  to  the  rate,  by  Art.  19, 

—  (cos^  X  -f-  sin-  x)  dx 


du  = 


sm''  X 
or,  since  cos^  x  -\-  sin^  x  =■  R~, 

R-dx 

du^^  — , 


sm-  X 

Therefore  the  rate  of  the  cotangent  is  equal  to  minus  the 
square  of  the  radius  into  the  rate  of  the  arc,  divided  by  the 
square  of  the  sine. 

If  u^  sec  X  = , 

cos  X 

then,  passing  to  the  rate, 

R^  sin  xdx 

du  ==■ . 

cos^  X 


28  AN  ELEMENTARY  TREATISE 

Therefore  the  rate  of  the  secant  is  the  square  of  the  radius 
into  the  sine  multiplied  by  the  rate  of  the  arc,  divided  by  the 
square  of  the  cosine. 

If  w=cosec;r  = , 


then  du  =  - 


sin;ir 
R^  cos  xdx 


sm"  X 

Therefore  the  rate  of  the  cosecant  is  equal  to  minus  the 
square  of  the  radius  into  the  cosine  multiplied  by  the  rate  of 
the  arc,  divided  by  the  square  of  the  sine. 

li  R  =  l,  then  for 

u  =  sin  X,  du  =  cos  xdx, 

u  =  cos  X,  du  =  —  sin  xdx, 

u  =  vers  X,  du  =  sin  xdx, 

u  =  tan  X,  dx 

du  = ,  etc. 

COS^  X 

If  u  =  s\nrx,  (1) 

by  assuming  rx  =  v,  then  will  sin  rx  =  sin  v, 

whence      dv  =  rdx  and  du  =  d  (sin  rx)  =  cos  vdv.  (2) 

Substituting  these  values  of  v  and  dv  m  (1),  then 

du  =  r  cos  rxdx. 

If  w  =  sin"x,  (3) 

by  assuming     sin  x^^v,  then  will  sin"  x  =  v^, 

whence  dv  =  cos  xdx  and  du  =  d  (sin"  x)  =  nv^-'^dv.  (4) 

Substituting  these  values  of  v  and  dv  m  (4)  gives 

du  =  n  sin""^  x  cos  xdx. 

If  w  =  sin"  ;ir  sin  r;r,  (5) 

then  du  =  d  (sin";r)  sin  rx  -{-  d  (sin  rx)  sin"  x; 

but,  as  shown,       d  (sin"  x)  =n  sin"~^  x  cos  xdx 

and  d  (sin  rjr)  ^r  cos  rxdx ; 

therefore 

du  =  (n  sin""^  x  cos  x  sin  rx  -\-  r  cos  rx  sin"  x)  dx, 

or         du  ==  sin""^  x  (n  cos  x  sin  rx  -\-  r  cos  rx  sin  x)  dx. 

By  use  of  these  equations,  the  rates  of  like  expressions  of 
the  cosine,  versed  sine,  tangent,  etc.,  can  be  determined. 


ON  VARIABLE  QUANTITIES  29 

32.    For  radius  R  and 

u  =  log  sin  X, 
by  passing  to  the  rate,  by  Arts.  27  and  31, 

cos  xdx 


R  sin  X 

the  rate  of  the  logarithmic  sine  of  the  arc  x. 

If  u==  log  cos  X, 

sin  xdx 
then  du  = 


R  cos  X 
the  rate  of  the  logarithmic  cosine  of  the  arc  x. 

If  u  =  log  vers  x 

or  its  equivalent,      u  =  log  {R  —  cos  x), 

sin  xdx 


then  du  = 


R  (R  —  cosx) 
smxdx 


or  du- 

R  vers  x 

the  rate  of  the  logarithmic  versed  sine  of  the  arc  x. 

If  u  =  log  tan  X 

R  sin  X 

or,  since  tan  x  = , 

cos  X 

R  sin.jr 
u  =  \og  ( ); 

cos  X 

then,  passing  to  the  rate,  by  Arts.  27  and  31,  and  reducing, 

cos^  xdx  +  sin^  xdx 

du  = 1 

R  sin  X  cos  x 

or,  since  cos^  x  -\-  sin^  x  =  R^, 

Rdx 

du  =  — ; , 

sin  X  cos  X 

the  rate  of  the  logarithmic  tangent  of  the  arc  x. 

R  cos  X 

If  u^  log  cot  X  =  log  — ; ; 

sin  X 


30  AN  ELEMENTARY  TREATISE 

—  (sin^  xdx  +  cos^  xdx)        R  cos  x 

then         du  = :       ^  ~       ; 

sm  x^  sin  X 

or,  since  sin^  x  +  cos^  x  =  R-,  by  reducing 

Rdx 


du 


sm  X  cos  X 

the  rate  of  the  logarithmic  cotangent  of  the  arc  x. 

R^ 
If  w  ^  log  sec  ;i;  =  loj 


then  du  ^= 


cos  X 
R^  sin  xdx  R^         sin  xdx 


cos  X       R  cos  X 

R  sin  X 

or,  since  =  tan  jr, 

cos  X 

tan  ;ir<f;ir 

du^= , 

R^ 

the  rate  of  the  logarithmic  secant  of  the  arc  x. 

In  like  manner  the  rate  of 

u  =  log  cosec  X 
cot  xdx 


will  be  found  to  be         du 

system  of  loga 

M  cos  xdx 


R' 
In  using  the  common  system  of  logarithms,  for 

u  =  log  sin  X,  du  = 


R  sin  X 

M  sin  xdx 
u  =  log  vers  x,  du  =  — -  ; 

R  vers  x 

M  sin  xdx 
u  =log  cos  X,  du  =  — 


R  cos  X 

MRdx 
u  =  log  tan  X,  du  =  —  ; 

sm  X  cos  X 

MRdx 
u  =  log  cot  X,  du  =  —  -  ,  etc. 

sm  X  cos  X 

33.     It  is  often  desirable  to  have  the  rate  of  the  arc  in 
terms  of  that  of  its  sine,  cosine,  tangent,  etc.,  and  for  this 


ON  VARIABLE  QUANTITIES  31 

purpose  the  following  expressions  are  employed:  namely, 
X  =  sin-^  u,  x^  cos-^  u,  x^  tan'^  u,  etc. ,  x  being  the  arc  and 
u  its  sine,  cosine,  etc. 

Let  X  =  sin"^  u , 

its  equivalent  being  u  =  sin  x, 

the  rate  of  which  is,  by  Art.  31, 

cos  xdx 


du 


whence  dx^ 


R 

Rdu 


cos  X 
or,  since  sin.;tr:^M,  consequently  cos.ar^  (7?^  —  u^Y^, 

Rdu 


dx^ 

the  rate  of  the  arc  in  terms  of  the  sine  and  its  rate. 

In  like  manner,  if        x=^  cos"^  u, 

Rdu 
it  will  be  found  that    dx=^ 


(R-2  —  U^)V. 
the  rate  of  the  arc  in  terms  of  the  cosine  and  its  rate. 

If  x^  vers"^  u, 

its  equivalent  being     u  =  vers  x  =  R  —  cos  x, 
the  rate  of  which  is,  by  Art.  31, 

sin  xdx 


du 


whence  dx  = 


R 
Rdu 


sm.ar 

Now  sin  .;r  :^  (R^  —  cos  x^)^'^^, 
but 

cos  .V  =^  R  —  vers  x  =  R  —  u ; 
therefore 

sin  x={R~—(R  —  uyy-  =  {2Ru  —  u-y^ ; 

Rdu 
hence  dx  = , 

(2Ru  —  u^y 

the  rate  of  the  arc  in  terms  of  the  versed  sine  and  its  rate. 


32  AN  ELEMENTARY  TREATISE 

If  x  =  tan"^  u, 

the  equivalent  being 

X  =  tan  X, 

the  rate  of  which  is,  by  Art.  31, 

R^dx 


du 


cos^  X 


cos^  xdu 

whence  dx  = ,  (1) 

R^ 

But  sec^  X :  R- : :  R^ :  cos^  x,  or,  since  sec^  x^R^  -\-  u-, 
R-  +  w^  :R2..]^2.  (,os2  ^^ 

R^ 


whence  cos-  x  ■ 


R^  -{-u^ 

Substituting  this  value  of  cos^  ;i;  in  ( 1 )  gives 

R~du 
dx  = , 

R^~  +  u^ 

the  rate  of  the  arc  in  terms  of  the  tangent  and  its  rate. 

In  like  manner,  if       x  =  cot~^  u, 

R^  du 
it  will  be  found  that  dx  =  • 


the  rate  of  the  arc  in  terms  of  the  cotangent  and  its  rate. 

34.  By  means  of  Maclaurin's  theorem,  sin  x,  cos  x,  etc.  can 
be  developed  in  terms  of  .*" :  thus,  if  i?  ^  1,  the  ratal  coefficients 
of  w  =  sin  X  are 

du                                                 d'u 
=  cos  X  =  —  sin  X 

dx  dx- 

d^u  d*u 


cos  X  =  sm  X 


dx^    .  dx' 

d^u 

=^  cos  X,  etc. 

dx^ 

Making  x^O,  then,  from  Art.  24, 

du  d^u  d^u 

A=0,  -—=1,  -7  =  0,  -—  =  —1, 

dx  ax-  ax" 


ON  VARIABLE  QUANTITIES  33 

=  0,  =1,  etc. 

dx^  dx^ 

By  substituting  these  values  in  Maclaurin's  theorem  the 
following  is  obtained: 

X           x^                    x^ 
u  =  s\r\  x^  —  — + ,  —  etc. 

1  2-3         2-3-4-5 

Proceeding  in  like  manner,  it  will  be  found  from  u  =  cos  x 

x^  X* 

that  M  =  1  —  —  + —  etc. 

1        2-3-4 

EXAMPLES 

Determine  the  rates  of  the  following: 

w  =  sin  ;ir^  u  =  tan  x^ 

u  =  sin  X  cos  X  w  =  cos  x  sin  x 

u  =  tan  X  log  cot  x  m  =  log  tan  x  -\-  log  cot  x 

u 


.s;  =  sin"^  2m  ( 1 — u)  .ar  =  cos"^ 

a  —  u 

b 
X  =  tan~^  — 

u 

Develop  x  =  sin"^  u. 

Vanishing  Fractions 

35.    A  vanishing  fraction  is  one  which  reduces  to  the  form 

0  .  .      .  - 

—  when  a  particular  value  is  given  to  the  variable.    Thus, 

c  {x  —  a) 

0 
reduces  to  —  when  x  =  a.   This,  it  will  be  seen,  is  owing  to  a 

factor  common  to  both  numerator  and   denominator   which 
reduces  to  zero  for  x^=a. 

Let  the  equation  be 


(1) 


34  AN  ELEMENTARY  TREATISE 

and  put  it  under  the  form 

{x  —  a)'"  {jir'""^  -f-  amx"^-~  -\-  a-  m  (m  —  \)x"^~^  -\ a'"^-'^Y 

U=:i ^ — 

{x  —  a)*'  {x"~'^  -\-  anx"^~'~  -j-  d-n  (n —  1)  x"~^  -\- a"-^}« 

Now  the  right-hand  portion  of  this  fraction  does  not  reduce 
to  zero  for  x^=^a;  therefore,  making  it  equal  to  R,  then 

(x  —  ay 

u  =  R- '-.  (2) 

{x  —  a)« 

Let    (x  —  a)''^P,    and    {x  —  a)*  =  Q,    both    of    which 
reduce  to  zero  for  x  =  a;  then 

RP 

(3) 


u  — 

'      Q    ' 

or 

Qu  = 

■  RP. 

Passing  to  the  rate 

Qdu  -f-  udQ  = 

■  RdP  +  PdR, 

butQ: 

=  OandP  =  0; 

therefore 

udQ  — 

-.RdP 
dP 

or 

u  =  R 

dQ  ' 

hence, 

RP 
since  u^ ,  (3) 

Q 

P 

u  =  R—-- 
Q 

dP 
=  R . 

dQ 

If  both  dP  and  dQ  reduce  to  zero  for  x^a,  then  by  pass- 
ing to  the  rate  again 

dP  d'P 

R =  R ; 

dQ  d-Q 

P  dP  d^P 

therefore  u  =  R  —  =:^  R =  R . 

Q  dQ  d'-Q 

Should  this  also  reduce  to  zero  for  x  ==  a,  by  continuing 
the  process  a  fraction  may  be  found  which  will  not  reduce  to 
zero  for  x  =  a,  and  thus  the  true  value  of  the  primitive  fraction 
will  be  obtained.* 


*  It  will  be  observed  that  the  process   employed  simply   eliminates  the 
vanishing  factor,  thus  giving  the  real  value  of  the  fraction. 


ON  VARIABLE  QUANTITIES  35 

Let  u={hx~  +  ex)  { ),  (4) 

X  —  a 

in  which  m  =  2,  m=1,  and  bx~ -\- ex    represents  R,  and  its 
rate  is 

d(x-  —  0")  2xdx 

u=(bx^  +  ex) ={bx~  +  ex) =  2bx^-\-2cx- 

d  {x  —  a)  dx 

or,  when  x^a,  u  =  2a^b  -{-2a^e.  (5) 

Multiplying  (bx^ -\- ex)  by  {x^  —  a^)  in  (4)  gives 

bx'^  -|-  ex^  —  a~bx-  —  a^cx 

u^ _ 

X  —  a 

Therefore 

(4bx^  -\-  3ex^  —  2a^bx  —  a^c)  dx 


dx 
4bx^  +  ^^•^"  —  2arbx  —  a^c, 
or,when;r  =  a,  u^2a^b-\-2a^e, 

the  same  result  as  (5). 

xn a"" 

If  u  = , 


X  —  a 

d  (x"-  —  a")      nx'^-^dx 

then  u  = = ^=nx^ 

d  (x  —  a)  dx 

or,  when  x  =  a,  u  =  na^~^. 

(;r2_  ^2)3/2 


If 


(x  —  a)^^^ 

by  squaring  and  then  taking  the  cube  root, 

(x^  —  a^) 

u'/'  =  — 

(x  —  a) 

2xdx 

Therefore  u~^^  = =  2x 

dx 

or,  when  x^a, 

y2iz —-2a  or  u=  {2a)^f^. 


36  AN  ELEMENTARY  TREATISE 


EXAMPLES 

u^ 

u 

(x^~2ax'-\-a^y^                  {2ax^- —  2d'y- 

{x'-  —  a^y^                       {x^ -\- a-x —  2a^y 

x^  —  2ax^  +  a^x                 a  —  x  -\-  a  log  x  —  a  log  a 

X-  —  a-                                  a — {2ax  —  x)"^ 

Det( 

irmine  the  value  oi  u  = ,  when  x  =  0. 

X  {x  -\-  c) 
Determine  the  value  of 

cos  X  —  sin  ;ir  -|-  1 


cos  X  -\-  sin  X —  1 


,  when  x  =  90°. 


In  some  cases  both  P  and  Q  become  infinite  for  a  particular 
value  of  the  variable :  thus 

tan;ir 


u 

cot  2x 

00 

becomes 

-when  ;ir  =  90°. 

00 

Now 

1                                       1 

tan  X  —             anc"  -  -  *  '^  ■ 

cot  X 

1    L,UL  i-A  , 

tan2;ir 

tan  X            tan  2x 

0 

therefore 

cot  2x            cot  X 

2dx 

o' 

Hence 

cos^  2x 

2  sin^  X 

li  

—  dx 

cos^  2x 

s\rr  X 

or,  since  sinjr==l  and  cos2;ir  =  — 1  when  ;ir  =  90°, 

u^  —  2. 

Sometimes  in  a  product  one  factor  becomes  zero  and  the 
other  infinite  for  a  particular  value  of  the  variable :  thus,  in 

1 
m:^(1 — ;i:)  tan  —  tt  x  (6) 


ON  VARIABLE  QUANTITIES  37 

1  1 

(1  — ;r)  =0  and  tan  — 7r:ir=co,  when  x=l  and  —  7r  =  90°. 
^         /  2  2 

1  1  .  , 

Since  tan  —  tt  ;ir  == ,  (6)  can  be  written  thus  : 

2  1 

cot  —  ttX 

2 

l—x 


1 

cot 77  X 

2 


then 


1 

dx  2  sin^  —  irx 

2 


1 

TT  dx  IT 

2 

~     i 

sin^  —  irx 
2 

1  1 

Therefore,  when  x=l  and  —  tt  =  90°,  since  sin  —  tt  jr  then 

2  2 

equals  1, 

2 

u  =  — . 


Of  the  difference  of  two  quantities,  both  sometimes  become 
infinite  for  a  particular  value  of  the  variable :  thus,  in 

^  =  -^— ^,  (7) 

X  —  1  log  ^ 

^  1  r     •  ' 

when  x=l,  both and. become  mfinite. 

X  —  1  log  X 

In  this  case  put  (7)  under  the  form 
X  log  X X  -\-  I 


(x —  1)  log;i; 

0 
which  becomes  —  when  x  =  1 ;  therefore 
0 

d  (x log X  —  X  -}-  I)  X log X 

u  = = 

d  (x  —  1 )  log  X  X  log  X  -\-  X  —  1 


38 


AN  ELEMENTARY  TREATISE 


This  also  becomes  —  when  x=l,  consequently 


dx  log  X 


xdx 


log  X  -\-  \ 


dx  log  X 


xdx 


+  dx 


log;r+l  +  l 


or,  when  jir  =  1,  since  log  1^0, 


1 

u  =  — . 
2 


EXAMPLE 


Determine  the  value  of 

u^ X  tan  X  — 


z  cos  X 


-,  when  x  =  90^. 


Curves  Referred  to  Rectangular  Coordinates 

36.     Signification  of  the  first  and  second  ratal  coefficients. 

Every   curve  or  line   referred   to   rectangular   coordinates 
may  generally  be  represented  by  the  equation 

y  =  f  {^) 

in  which  x  represents  any  abscissa,   as  AB,  and  y  the   cor- 
responding ordinate  BP,  of  the  curve  CPD  (see  figure). 


Fig.    7 

Draw  TT'  tangent  to  the  curve  CPD  at  P,  and  with  radius 
unity  draw  the  arc  EH,  also  draw  EK  tangent  to  EH.  Then 
the  angle  T'TX  will  be  the  angle  of  tangency,  so  called,  and 


ON  VARIABLE  QUANTITIES 


39 


EK  its  tangent,  which  is  designated  by  t.    Now  let  dx  be  re- 
presented by  PG,  and  dy  by  GF  (see  Art.  30)  ;  then  will 

dx:dy::  TE :  EK 

or,  since   TE^l,  and  EK^t,  by  representing  the  rate  of 
/  (x)  by  /'  (X) 

dx:  dy::  I:  t, 

dy 


whence 


^  = 


dx 


Passing  to  the  rate  and  representing  the  rate  oi  f  (x)  by 

dt        d^y 

=  — ^  =  /-(x). 

dx        dx^ 

Hence  the  first  ratal  coefficient  of  the  equation  of  a  curve 
represents  the  tangent  of  the  angle  of  tangency  of  any  point  of 
the  curve  and  the  second  ratal  coefficient  represents  the  rate  of 
variation  of  the  tangent  of  the  angle  of  tangency. 

Z7.  Of  a  curve  concave  to  the  axis  of  X  (Fig.  8),  the 
angle  of  tangency  and  consequently  its  tangent  t  decrease  as 
the  ordinate  increases,  as  is  clearly  shown  by  the  tangents  TP 
and  TP'  of  the  curve  CPP'D. 


This,  it  will  be  observed,  is  also  true  of  the  curve  C'QQ'D' , 
as  indicated  by  the  tangents  TQ  and  T'Q';  but,  lying  below 
the  axis  of  X,  the  angle  of  tangency  and  its  tangent,  as  well  as 
the  ordinate,  are  negative,  while  those  above  the  axis  are 
positive. 

Now,  since  the  rate  of  a  positive  decreasing  function  is 
negative  and  that  of  a  negative  decreasing  function  is  positive, 


40 


AN  ELEMENTARY  TREATISE 


the  rate  of  the  tangent  of  the  angle  of  tangency  of  the  curve 
CPP'D  is  negative,  while  that  of  the  curve  C'QQ'D'  is  posi- 
tive. Therefore,  since  t  represents  the  tangent  of  the  angle 
of  tangency,  when  a  curve  is  concave  to  the  axis  of  x  and  its 
ordinate  is  positive,  dt,  and  consequently  the  second  ratal  co- 
efficient of  the  equation  of  the  curve,  are  negative,  but  positive 
when  the  ordinate  is  negative. 

If  the  curve  is  convex  to  the  axis  of  X  (Fig.  9),  the  angle 
of  tangency,  and  consequently  its  tangent  t,  increase  as  the 
ordinate  increases,  as  is  shown  by  the  tangents  TP  and  T'P' 
of  the  curve  CPP'D. 


This  is  also  true  of  the  curve  C'QQ'D' ,  as  indicated  by 
the  tangents  TQ  and  T'Q' ;  but  as  they  lie  below  the  axis  of  X, 
the  angle  of  tangency  and  its  tangent,  as  well  as  the  ordinate, 
are  negative. 

Therefore,  since  the  rate  of  a  positive  increasing  function  is 
positive,  and  that  of  a  negative  increasing  function  is  negative, 
the  rate  of  the  tangent  of  the  angle  of  tangency  for  any  point  of 
the  curve  CPP'D  is  positive,  while  that  of  the  tangency  for  any 
point  of  the  curve  C'QQ'D'  is  negative. 

Hence  when  a  curve  is  convex  to  the  axis  of  X,  and  its 
ordinate  is  positive  and  increasing,  dt,  and  consequently  the 
second  ratal  coefficient  of  the  equation  of  the  curve,  are  posi- 
tive, hut  negative  when  the  ordinate  is  negative. 

From  what  precedes,  the  following  conclusion  is  evident: 


ON  VARIABLE  QUANTITIES  41 

When  the  second  ratal  coefficient  of  the  equation  of  a  curve 
is  negatizre,  the  curve  is  either  concave  to  and  above  the  axis  of 
X  or  convex  to  and  below  it;  but  when  the  second  ratal  co- 
efficient is  positive,  the  curve  is  either  concave  to  and  below  the 
axis  or  convex  to  and  above  it.    . 

Sometimes  a  particular  value  of  x,  as  ;r  =  a,  will  make 

d^'y 

=  0.    In  this  case,  substitute  a  ±v  ior  x;  then,  for  a  small 

dx'- 

d^y 

value  of  V,  if  and  the  ordinate  corresponding  to  x=^a 

dx~ 

have  contrary  signs,  the  curve  is  concave  to  the  axis  of  X  at  a 
point  in  the  curve  whose  abscissa  is  x^a,  but  if  like  signs, 
convex. 

E.  g.,  let  the  equation  of  the  curve  be 

y  =  x^  —  Sx^^  AOx-  —  mx  +  58. 

Passing  to  the  rate  twice, 

d^y 

-^  =  20.^^  —  60^^  +  80,  (1) 

dx^ 

d'y 
in  which  =  0  when  x  =  2;  therefore,  substituting  2  =t  ^^ 

dx'-^ 

for  JIT  in  (1)  gives 

d'y 

=  60?v2  ±:  20z/3 

dx' 

d^y 
or  -^  =  20v'  (3±v), 

dx" 

which  is  positive  for  any  value  oi  v  <i  3.  Hence,  since  3;  =  10 
when  x^^2,  the  curve  is  convex  to  the  axis  of  X  at  the  point 
whose  abscissa  is  ;ir  =  2. 

Determine  whether  the  curve  whose  equation  is 

3;  =  5  -{-  4x  —  x~, 

is  concave  or  convex  to  the  axis  of  X. 

Ratal  Equations  of  Lines 

38.  A  ratal  equation  of  a  line  is  one  which  shows  the  rela- 
tion between  the  coordinates  and  their  rates,  and,  being  inde- 
pendent of  the  values  of  the  constants  which  enter  the  primitive 


42  AN  ELEMENTARY  TREATISE 

equation,  determines  the  general  nature  of  the  Hne  without 
regard  to  its  magnitude. 

First  take  the  general  equation  of  lines  of  the  first  order 

dy 

whence  ^  a, 

dx 

a  result  which  is  the  same  for  all  values  of  b.  This  equation 
represents  the  tangent  of  the  angle  of  tangency  (see  Art.  36) 
and  is  the  first  ratal  equation  of  lines  of  the  first  order. 

Passing  to  the  rate  again 

dy  d-y 

d-^=^0  or  -^  =  0, 
dx  dx- 

an  equation  entirely  independent  of  the  values  of  a  and  b  and 
consequently  equally  applicable  to  every  line  of  the  first  order 
which  can  be  drawn  in  the  plane  of  the  coordinate  axes.  It  is 
called  the  general  ratal  equation  of  lines  of  the  first  order. 

d~y 

The  equation =  0  shows  that  the  tangent  of  the  angle 

dx'^ 

of  tangency  has  no  variation  (see  Art.  Z7)  ;  hence  every  line 
of  the  first  order  must  necessarily  be  a  straight  line. 

39.     In  the  general  equation  of  lines  of  the  second  order 

y^  =  ax~  -{-  bx  -\-  c,  ( 1 ) 

passing  to  the  rate  thrice  will  give 

2ydy 
-^—^  =  2ax-{-b  (2) 

dx 
dy^        yd^y 

-^Jr^-^  =  a.  (3) 

dx'  dx- 

dyd-y         yd^y 

-^—^  +  -^^—^  =  0;  (4) 

dx^  dx^ 

equations  (2),  (3),  and  (4)  are  respectively  the  first,  second, 
and  general  ratal  equations  of  lines  of  the  second  order. 

When  the  origin  of  coordinates  is  at  the  vertex  of  the  trans- 
verse axis,  the  general  equation  of  lines  of  the  second  order  is 

y^  =r=  ax"^  -\-  bx.  (5) 


ON  VARIABLE  QUANTITIES 


43 


Passing  to  the  rate  twice 

2ydy  =  2axdx  -\-  bdx  (6) 

and  dy^  -\-  yd^y  =  adx"^.  (7) 

Eliminating  a  and  &  in  (5)  by  means  of  (6)  and  (7) 
y-dx^  +  x^dy-  -\-  x'^yd'^y  —  2xydxdy  =  0, 
a  general  ratal  equation  of  lines  of  the  second  order,  when  the 
origin  of  the  coordinates  is  at  the  vertex  of  the  transverse  axis, 
differing  from  (4)  on  account  of  passing  to  the  rate  but  twice. 

Determine  the  general  ratal  equations  of  the  circle,  para- 
bola, and  ellipse. 

Tangents  and  Normals 
40.    li  X  and  y  represent  the  coordinates  of  any  point  of  the 
curve  APC,  and  z  the  corresponding  arc  AP,  Art.  30,  dx  will 

be  represented  by 
DP,  dy  by  DE,  and 
dz  by  PE,  a  tan- 
gent to  the  curve  at 
the  point  P ;  also 
iO  TR  represents  the 
subtangent,  TP  the 
tangent,    RN    the 

-p  -^ p nh nZ    subnormal,  and  PN 

the  normal,  each  of 
which  are  obtained  as  follows : 

DE:DP::PR:TR; 

dy:  dx 


that  is. 


or 


TR 


y.TR, 
ydx 


that  is, 


or 


that  is. 


or 


dy 
TP^-  =  PR^-  +  TR^- ; 

TP^  =  3,2  ^  £ 

dy- 

TP  =  —  (dx'  -\-  dy^-y\ 
dy 

DP:\DE::PR:RN; 

dx:dy\\y:  RN 

ydy 


(1) 


(2) 


RN=' 


dx 


(3) 


44  AN  ELEMENTARY  TREATISE 

PN^  =  PR^-\-RN^; 
that  is,  PN^  =  3,2  _p  3,2 

or  PN  =  -^  {dx- ^  dy'^y-.  (4) 

dx 

1.  Hence  the  length  of  the  subtangent  to  any  point  of  a 
curve  is  equal  to  the  ordinate  into  the  rate  of  the  abscissa 
divided  by  the  rate  of  the  ordinate. 

2.  The  length  of  the  tangent  to  any  point  of  a  curve  is 
equal  to  the  ordinate  divided  by  the  rate  of  the  abscissa,  into 
the  square  root  of  the  sum  of  the  squares  of  the  rates  of  the 
abscissa  and  ordinate. 

3.  The  length  of  the  subnormal  to  any  point  of  a  curve  is 
equal  to  the  ordinate  into  its  rate  divided  by  the  rate  of  the 
abscissa. 

4.  The  length  of  the  normal  to  any  point  of  a  curve  is  equal 
to  the  ordinate  divided  by  the  rate  of  the  abscissa,  into  the 
square  root  of  the  sum,  of  the  squares  of  the  rates  of  the 
abscissa  and  ordinate. 

The  tangent  TP  may  also  be  obtained  thus : 

DE:PE::PR:TP ; 

that  is,  dy.dsr.y:  TP 

dz 
or  TP^y--.  (5) 

dy 

Likewise,  for  the  normal  PN, 

DP:PE::PR:PN; 

that  is,  dx:ds::y:  PN 

dz 

or  PN  =  y .  (6) 

dx 

In  the  application  of  these  formulas  to  any  particular  curve, 

dx  dy  _ 

the  value  of  or  ,  obtained  from  the  equation  of  the 

dy  dx 

curve  by  passing  to  the  rate,  must  be  substituted  in  each  of 
them.  The  result  will  be  true  for  all  points  of  the  curve ;  then, 
by  substituting  therein  the  values  of  x  and  y  for  any  particular 


ON  VARIABLE  QUANTITIES  45 

point  of  the  curve,  we  can  find  the  value  of  the  subtangent, 
tangent,  subnormal,  and  normal  for  that  point. 

41.  To  apply  the  formulas  of  the  preceding  articles  to  lines 
of  the  second  order,  whose  general  equation  is 

y~  =  ax^  -\-  bx  -\-  c, 
and  its  rate 

dy         2ax  -(-  ^  2a;ir  -f-  h 

dx  2y  2  {ax^  -\-  bx  -{-  c)''^ 

dy 

Substituting  the  value  of  in  formulas  (1),  (2),  (3), 

dx 
and  (4)  of  Art.  40,  will  give 

dx        2{ax^  -\-  bx  -\-  c) 


TR  =  y 


dy  2ax  -\-  b 


TP  =  —{dx'--^  dy^ )  ^/^  = 
dy 

ax^  -{-  bx  -\-  c 

^{ax^-  -^bx^c  +4  ( -—-)! 

lax  -\-  b 

dy        2ax  4-  b 
RN  =  y—^  = 


dx 

y  1 

PN  = (dx""  +  dy^y''=^/{ax^  -j- bx  +  c -\-  —{2ax  +  by). 

dx  4 

By  giving  proper  values  to  a,  b,  and  c,  these  formulas  will 
become  applicable  to  any  line  of  the  second  order. 

In  the  case  of  the  parabola,  a  =  0,  b  =  2p,  and  c  ==  0 ; 
therefore  TR  =  2x 

TP  =  {2px  ^  Ax^y^ 
RN  =  p 

PN  =  (2px -{- p-y. 

EXAMPLES 

The  major  axis  of  an  ellipse  is  40  inches  and  the  minor 
axis  is  20  inches.  What  are  the  lengths  of  the  tangent,  sub- 
tangent,  normal,  and  subnormal? 

The  transverse  axis  of  a  hyperbola  is  6  inches  and  the 
conjugate  axis  is  4  inches.  What  is  the  length  of  the  sub- 
normal corresponding  to  an  abscissa  of  9  inches,  the  equation 


46 


AN  ELEMENTARY  TREATISE 


being  y^ 


A' 


52? 


The  Cycloid 
42.     If  a  circle,  NPG,  be  rolled  along  a  straight  line  AC, 
any  point  P  of  its  circumference  will  describe  a  curve  called 
a  cycloid.    The  circle  NPG  is  called  the  generating  circle  and 
the  point  P  is  called  the  generating  point. 

The  line  AC  is  equal  to  the  circumference  of  the  generat- 
ing circle  and  is 
called  the  base  of 
the  cycloid,  and  the 
line  BD,  drawn 
perpendicular  to  it 
at  its  middle  point, 
is  called  the  axis 
of  the  cycloid  and 
is  equal  to  the  di- 
TA  n        N  D  C   ameter  of  the  gen- 

erating circle. 

In  determining 
the  equation  of  the  cycloid,  take  the  origin  of  the  coordinates 
at  A  and  suppose  that  the  generating  point  has  described  the 
arc  AP ;  then  if  N  be  the  point  at  which  the  generating  circle 
touches  the  base,  AN  will  be  equal  to  the  arc  NP. 

Draw  NO,  the  diameter  of  the  generating  circle,  PR  perpen- 
dicular to  the  base,  and  PH  parallel  to  it ;  then  PR  will  be  equal 
to  NH  which  is  the  versed  sine  of  the  arc  NP. 

Let  NO  =  2r,  AR  =  x,  and  PR^HN  =  y\  then 


RN  =  PH=  {NH-HGy-=  (y2r  —  yy-=  (Zry  —  y^)^-; 

also  X  =AR  =  AN  —  RN  =  arc  NP  —  PH. 

Therefore,  since  NP  is  the  arc  whose  versed  sine  is  NH  or 
y  (that  is,  NP  =  vers  ''^y), 

X  =  vers  "^3;  —  ( 2ry  —  y~)V2  ( 1  -j 

which  is  the  transcendental  equation  of  the  cycloid. 

rdy 

The  rate  of  vers  "^3^  is and  that  of  {2ry — y^)"^^ 

{2ry—y^)'^ 

rdy  —  ydy 
is ;  therefore 

{Iry  —  y^-y^ 


dx  ^ 


ON  VARIABLE  QUANTITIES  47 

rdy  rdy  —  ydy 


{2ry  —  y~y^^         {2ry  —  y^)^ 


ydy 
or  a?;r  = ,  (2) 

{^2ry  —  y^y- 

which  is  the  ratal  equation  of  the  cycloid. 

43.  Dividing  both  members  of  (2)  of  the  preceding  article 
by  dy,  gives 

dx  y 

dy        {2ry  —  y'^y^ 

dx 

then,  by  substituting  this  value  of  in  formulas   (1),  (2), 

dy 

(3),  and  (4)  of  Art.  40,  the  values  of  subtangent,  tangent, 
subnormal,  and  normal  for  any  point  of  the  cycloid  are  as 
follows : 

TR  = 

(^2ry  —  y^y^ 

y  (2ryy^ 
TP=  

(2ry  —  Z)'/^ 

RN=  {2ry  —  y~y 
PN=^{2ry). 

The  Logarithmic  Curve 

44.  The  logarithmic  curve  takes  its  name  from  the  pro- 
perty that,  when  referred  to  rectangular  axes,  any  abscissa  is 
equal  to  the  logarithm  of  the  corresponding  ordinate;  hence 
the  equation  of  the  curve  is 

X  =  log  y. 

If  a  represents  the  base  of  one  system  of  logarithms,  and  b 
that  of  another,  then 

gx  :—-y    ^jj(J    ^a;  __  y  . 

from  which  it  is  evident  that  for  every  different  base  the  same 
value  of  x  will  give  a  different  value  for  3; ;  that  is,  every  dif- 
ferent logarithmic  base  will  give  a  different  logarithmic  curve. 

If  we  make  x^=0,  y  =  1 ;  therefore,  since  this  relation  is 
independent  of  the  base  of  the  system,  it  follows  that  every 
logarithmic  curve  will  intersect  the  axis  of  ordinates  at  a  dis- 
tance from  the  origin  equal  to  unity. 

From  a'^  =  y 

the  curve  can  be  described  by  points  even  without  the  aid  of 
a  table  of  logarithms ;  thus 


^A^  ELEMENTARY  TREATISE 


etc. 


x=l,     y  =  a; 
x  =  — 1,     y^a-'^ 

Then    if    the    origin 

is   at  A    (see  Fig.    12), 

2  BCD  will  be  the  curve. 

Resuming  the   equa- 
tion of  the  curve, 

2  -^  =  log  3' ; 

then,  if  M  represents  the  modulus  of  the  system,  the  rate  is 

Mdy 


dx  =  - 
dx 


y 

M 


or 


dy 


y 


dx 


Substituting  this  value  of  in  formula  (1)  of  Art.  40 

dy 
gives 

dx 

TR  =  y =  M. 

dy 

Hence  the  subtangent 
of  the  logarithmic  curve 
is  constant  and  equal  to 
the  modulus  of  the  sys- 
tem, in  which  the  loga- 
rithms are  taken  (see 
Fig.   13). 

In  the  Naperian  sys- 
tem M  =  \;  consequently 
the  subtangent  of  the 
curve  in  this  system  is 
equal  to  unity  or  AB. 

TAR  X        If  the  value  of  — 

dy 

^§-    '^  be  substituted  in  formu- 

las (2),  (3),  and  (4)  of  Art.  40,  then 


ON  VARIABLE  QUANTITIES 


49 


RN  = 


f 


M 


M 


Asymptotes 
45.  An   asymptote   is   a   right   line   which   continually   ap- 
proaches a  curve  and  becomes  tangent  to  it  only  at  an  infinite 
distance  from  the  origin  of  the  coordinates. 


Let  A  be  the  origin  of  the  coordinates  (see  Fig.  14),  and 
let  TE  be  tangent  to  the  curve  at  P ;  then,  since  the  subtangent 

dx 

TR  is  equal  to  y and  the  abscissa  AR  =  x, 

dy 

AT^TR  —  AR 


or 


dx 

AT  =  y —  X. 

dy 


(1) 


Also  DP:DE::AT:AB 

or,  since  DP  is  represented  by  dx,  DE  by  dy,  and 

dx 


AT  =  y 
dx:dy::y 


dy 
dx 


■  X 


dy 


x:AB, 


50  AN  ELEMENTARY  TREATISE 

dy 

whence  AB  =  y  —  x .  (2) 

dx 

If,  when  X  and  y  become  infinite,  both  of  the  expressions 
(1)  and  (2)  also  become  infinite,  it  is  evident  the  curve  has 
no  asymptote;  but  if  either  one  or  both  of  the  expressions 
reduce  to  a  finite  quantity,  it  may  be  inferred  that  the  curve 
has  an  asymptote. 

If  both  expressions  are  finite,  the  asymptote  will  be  in- 
clined to  both  tlie  coordinate  axes ;  if  one  becomes  finite  and 
the  other  infinite,  the  asymptote  will  be  parallel  to  one  of  the 
coordinate  axes;  if  both  become  zero,  the  asymptote  will  pass 
through  the  origin  of  coordinates. 

The  general  equation  of  lines  of  the  second  order  is 
y^  =  ax^  -\-  bx  -\-  c. 

Passing  to  the  rate, 

2ydy  =  (2ax  -\-  b)  dx, 

dx  2y^ 

whence  y = 

dy  2ax  -f-  b 

or,  substituting  for  2y^  its  value, 

dx         2ax^  -(-  2bx  -\-  2c 


y 


dy  2ax  ~\-  b 

it  will  be  found  that 

dy  2ax^  -\-  bx 


dx         2  (ax^  -\-  bx  -\-  cy^ 

dx  dy 

Substituting  these  values  of  y and  x  — \ —  in  ( 1 )   and 

dy  dx 

(2)  gives 

2ax'^  -\-  2bx  -\-  2c  bx  -\-  2c 

AT  = ■ —  x  = ,  (3) 

2ax  +  b  2ax  -j-  b 

AB=  {ax-  -^  bx^cy  — 

2ax-  -\-  bx  bx  -\-  2c 

2  (ax^ -\- bx -]- cy~  2  (ax- -\- bx  ^  cy 


(4) 


ON  VARIABLE  QUANTITIES  51 

The  equation  of  the  parabola  is 
y^  =  2px ; 
hence  in  y""  =  ax^  -\- bx -\- c,  a  =  0,  b  =  2p,  and  c  =  0 ;  there- 
fore (3)  and  (4)  become 

2px 

AT  = =  x 

2p 

2px  1 

and  AB=  =-^{2px). 

2\/{2px)         2 

Making  x  infinite,  these  equations  become   infinite   also; 
therefore  the  parabola  has  no  asymptote. 

In  the  equation  of  the  circle  and  ellipse,  a  in 

is  negative;  consequently  AB  becomes  imaginary  when  x  is 
infinite;  therefore  neither  the  circle  nor  the  ellipse  has  an 
asymptote. 

The  equation  of  the  hyperbola  is 

-y-  = x^  —  B- ; 

A^ 

hence  &  =  0  in  y^  ^  ax^ -\- bx -\- c ;  therefore  (3)  and  (4) 
become 

2c  c 

AT  = 


and  AB 


2ax        ax 

2c  c 


2{ax^-^cy-       {ax'--\-cy 


When  X  is  infinite  both  of  these  equations  become  equal  to 
zero ;  hence  the  hyperbola  has  asymptotes,  one  to  either  branch 
of  the  curve,  both  of  which  pass  through  the  origin  of  coor- 
dinates. 

The  equation  of  the  logarithmic  curve  is 
X  =  log  y. 


52 


AN  ELEMENTARY  TREATISE 


Passing  to  the  rate 


dx 


Mdy 


y 


whence 


dx 


dy 


y 


M  and  x = 


xy 
dy  dx         M  ' 

Substituting  these  values  in  (1)  and  (2)  gives 
dx 


AT  =  y 


dy 


=  M  —  X 


and 


AB  =  y 


dy 


xy 


=  y 


dx       ^         M 

When  y  =  0,  X  is  negative  and  infinite ;  but  when  x  is  nega- 
tive and  infinite  (5)  and  (6)  become 

AT  =ao  and  AB  =  0; 

hence  the  axis  of  abscissas  of  the  logarithmic  curve  is  an 
asymptote  to  that  branch  of  the  curve  which  lies  on  the  left 
of  the  origin  of  the  coordinates.  (See  Fig.  13  in  Art.  44.) 

Rates  of  Arc,  Area,  Surface,  and  Volume 
OF  Revolution 
46.     For  the  rate  of  an  arc,  see  Art.  30,  in  which  the 
formula  thereof  will  be  found,  viz., 

d2=(dx^-\-dy^y'^, 
z  being  the  arc,  x  the  abscissa,  and  3*  the  ordinate. 

Hence  the  rate  of  an  arc  is  equal  to  the  square  root  of  the 
sum  of  the  square  of  the  rates  of  the  coordinates. 

47.  Let  X  represent 
the  abscissa  AD  (see 
Fig.  15),  y  the  ordinate 
DP,  and  let  PG  or  DH 
be  represented  by  dx, 
the  rate  of  ^;  then  the 
rate  of  increase  of  the 
area  APD  will  be  repre- 
sented by  ydx.  There- 
X  fore,  if  the  area  of  APD 
be  represented  by  A,  then 
dA  ^=  ydx. 


ON  VARIABLE  QUANTITIES 


53 


Hence  the  rate  of  the  area  of  a  segment  of  a  curve  is  equal 
to  the  ordinate  into  the  rate  of  the  abscissa. 

48.  Let  X  represent  the  abscissa  AD,  y  the  ordinate  DP, 

2  the  arc  AP,  and  S 
the  surface  of  revolu- 
tion made  by  the  arc 
AP  in  revolving  round 
AD  or  the  axis  of  X ; 
then   the   point   P   of 

_  the  curve  APC  will 
describe  a  circle  whose 
radius  is  y  and  conse- 
quently its  circumfer- 
ence 2 Try.  Now  it  is 
J-  ^  >^  /  evident  that  if  2iry  be 

•■'g-    '-^  P'^==^^  multiplied    by    dz 

(which  equals  PT)   the  rate  at  which  z,  or  the  arc  APC,  is 
increasing  at  P,  then 

dS^2  nydz, 

or,  substituting  for  dz  its  value  from  Art.  46, 

dS  =  2  Try  idx-  +  dy^y^. 

Hence  the  rate  of  the  surface  of  the  volume  of  revolution 
of  an  arc  of  a  curve  is  equal  to  the  circumference  of  a  circle 
whose  radius  is  the  ordinate  of  the  arc,  multiplied  by  the  rate 
of  the  arc. 

49.  Let  X  represent  any  abscissa,  as  AD,  y  the  correspond- 

P       G 


54  AN  ELEMENTARY  TREATISE 

ing  ordinate  DP,  and  let  dx  be  represented  by  DH  =  PG  ;  then 
dx  multiplied  by  the  area  of  the  circle  described  by  the  point  P, 
in  revolving  round  the  axis  of  X,  is  equal  to  the  rate  at  which 
the  volume  of  revolution  is  increasing  when  the  arc  is  AP,  and 
consequently  the  ordinate  is  DP.  Therefore,  since  DP  =  y, 
TT'f-  is  equal  to  the  area  of  the  circle  described  by  the  point  P ; 
consequently,  if  V  represents  the  volume  of  revolution  gen- 
erated by  the  arc  AP  in  revolving  round  the  axis  of  X,  then, 
since  PG  =  dx, 

dV  =  TT  y^dx. 

Hence  the  rate  of  the  volume  of  revolution  of  an  arc  of  a 
curve  is  equal  to  the  area  of  the  circle  whose  radius  is  the 
ordinate  of  the  arc,  into  the  rate  of  the  abscissa. 

EXAMPLES 

Determine  the  rate  of  an  arc  of  the  circle  whose  equation  is 

Passing  to  the  rate, 

x'dx^ 
ydy  =  —  xdx  or  dy~  = , 

but,  by  Art.  46,  ds  =  (dx^ -\-  dy^ y^; 

therefore 

x-dx'^                         dx 
dz={dx^~  + y-  or  dz=: {x^^-y^y. 

y-  y 

But,  since  y=  (r-  —  x^)^'^-  or  {x^  -\-  y^)''-^r,  then 

rdx 
dz  = . 

(^r^-  —  x'-y 

Determine  the  rate  of  the  area  of  the  parabola ;  also  the  rate 
of  the  surface  and  volume  of  revolution  of  the  hyperbola. 

Radius  of  Curvature 

50.  Of  curves  tangent  to  each  other  and  having  a  common 
tangent  line  at  the  point  of  contact,  the  one  which  departs 
most  rapidly  from  the  tangent  line  is  said  to  have  the  greatest 
curvature.  The  curvature  of  a  circle  is  measured  by  the  angle 
formed  by  the  radii  drawn  through  the  extremities  of  an  arc 
of  a  given  length. 

Let  R  and  R'  represent  the  radii  of  two  circles,  a  the  length 
of  a  given  arc  measured  on  the  circumference  of  each,  c  the 
angle  formed  by  the  radii  drawn  through  the  extremities  of  the 


ON  VARIABLE  QUANTITIES  55 

arc  of  the  one  having  radius  R,  and  c'  the  angle  similarly 
formed  by  the  radii  of  the  one  having  radius  R' ;  then 
27ri?:360°::o:c  and  2  tt  R' :  ?>60°  : :  a :  c' , 

360°  a  360°  a 

vi^hence  c  = and  c' 


ZttR  2TrR' 

360° a     360° a  1        1 

therefore  c:c':: : or  c:c'::  —— :  ——. 

2TrR      2-kR\  R      R' 

Hence  the  curvature  in  two  different  circles  varies  inversely 
as  their  radii. 

Make  TNG  a  tangent  line  to  the  curve  ANC,  touching  at 
the  point  A'',  and  NM  a  normal  line  thereto  (see  figure)  ;  then 

G 


the  circumference  of  every  circle  having  its  center  in  NM, 
which  may  be  described  through  the  point  N,  will  touch  at  A^ 
both  the  curve  ANC  and  the  tangent  line  TNG. 

Now  it  is  obvious  that  the  circumference  of  any  such  circle 
which  has  a  greater  curvature  than  the  curve  ANC  will  depart 
more  rapidly  from  the  tangent  line  than  ANC  and  consequently 
will  fall  wholly  within  ANC ;  but  any  circumference  which  has 
a  less  curvature  than  ANC  will  depart  less  rapidly  from  the 
tangent  line  than  ANC  and  consequently  will  fall  between  it 
and  TNG.  Hence,  since  there  may  be  circumferences  of  both 
less  and  greater  curvature  than  ANC,  it  follows  that,  with  a 
center  in  the  line  NM,  a  circumference  may  be  described 
through  the  point  N  whose  curvature  will  correspond  with  that 
of  the  curve  ANC  at  N — that  is,  which  will  depart  from  the 
tangent  line  at  N  at  the  same  rate  as  the  curve  ANC. 

The  circle,  the  curvature  of  whose  circumference  corre- 
sponds  with   that   of   any   curve  at  any   point,   is   called   the 


56 


AN  ELEMENTARY  TREATISE 


osculatory  circle  or  circle  of  curvature,  and  its  radius  the  radius 
of  curvature  of  the  curve. 

51.  Let  the  curves  BNC  and  DNE  be  tangent  to  each  other 
at  N,  and  draw  TNG  a  tangent  line  to  both,  touching  at  N  (see 
figure);  also  let  y  =  f  {x)  represent  the  curve  BNC  and 
y'=f{x')  that  of  DNE.  Draw  the  ordinate  LN ;  then, 
since  LN  is  common  to  both  curves,  y^^y'  for  the  point  N ; 
also,  since  the  angle  LTN  and  consequently  the  tangent  of  the 
angle  of  tangency  for  the  point  A^  are  common  to  both  curves, 

dy        dy'  _  _  dy        dy' 

= .     These  two  conditions,  y=^y'  and  ^ , 

dx        dx'  dx       dx' 

existing,  the  curves  are  said  to  have  a  contact  of  the  first  order. 

If  at  the  point  N  the  second  ratal  coefficients  of  the  equa- 

d'^y       d^y' 

tions  of  the  two  curves  are  also  equal  (that  is,  if = ) 

dx"       dx^^ 


c 


there  will  be  a  so-called  contact  of  the  second  order.     This  is 

evident  since  either  or  is  the  same  as  the  rate  of 

dx^         dx'^ 

variation  of  the  tangent  of  the  angle  LTN  (the  angle  of 
tangency)  ;  consequently  both  curves  depart  from  the  tangent 
line  TNG  at  the  same  rate. 

If,  in  addition,  the  third  ratal  coefficients  of  the  equations 

d^y        d^y' 

of  the  curves  are  equal  (that  is,  if == )  the  curves  will 

dx^      dx'^ 

have  a  contact  of  the  third  order,  and  so  on  for  any  order  of 
of  contact. 

Now  if  BNC  be  given  in  species,  magnitude,  and  position, 


ON  VARIABLE  QUANTITIES  57 

and  DNE  in  species  only,  then  the  constants  which  enter 
y^=f  {x)  will  be  fixed  and  determinate,  while  those  which 
enter  y'  =f  {x')  will  be  entirely  arbitrary,  and  therefore  their 
values  may  be  made  to  answer  as  many  independent  condi- 
tions as  there  are  constants.  Hence  for  a  contact  of  the  first 
order,  y^f  {x)  must  contain  at  least  two  constants;  for  a 
contact  of  the  second  order,  three  constants ;  for  a  contact  of 
the  third  order,  four  constants,  and  so  on. 

In  the  most  general  equation  of  the  straight  line,  which  is 

y^ax  -]rh, 

there  are  two  constants,  a  and  h ;  therefore  the  straight  line 
can  have  only  a  contact  of  the  first  order. 

The  most  general  equation  of  the  circle  is 

(^y  —  by  =  R-—  {x  —  ay, 

which  contains  three  constants,  a,  b,  and  R ;  therefore  the 
circle  can  have  a  contact  of  the  second  order. 

In  the  general  equation  of  the  parabola, 

{ (y  —  h)  cos  V  —  {x  —  a)  sin  vY  = 
^P  {(y  —  ^)  sin  V  -{-  (x  —  a)  cos  v], 

there  are  four  constants,  a,  b,  v,  and  p ;  therefore  the  parabola 
can  have  a  contact  of  the  third  order. 

In  the  general  equation  of  the  ellipse  or  hyperbola  there 
are  five  constants ;  therefore  either  the  ellipse  or  hyperbola 
can  have  a  contact  of  the  fourth  order. 

The  curve  which  has  a  higher  order  of  contact  with  a  given 
curve  than  can  be  found  for  any  other  curve  of  the  same 
species  is  called  the  osculatrix  of  that  species. 

52.    The  general  equation  of  the  circle  is 

{y  —  by  =  R''—{x  —  ay.  (1) 

Passing  to  the  rate  twice,  under  the  supposition  that  neither 
X  nor  y  varies  uniformly — that  is,  that  neither  dx  nor  dy  is 
constant — then 

(y  —  b)  dy  =  —  (x  —  a)  dx 

and  (y  —  b)d''y-\-dy''  =  —(x  —  a)d^x  —  dx^. 

From  these  two  equations  the  following  are  found: 

(dx^  +  dy^)  dx 

dxd^y  —  dyd^x 


58  AN  ELEMENTARY  TREATISE 

{dx^  -\-  dy^)  dy 

and  X  —  a  = . 

dxd^y  —  dyd'X 

Substituting  these  values  of  y  —  b  and  x  —  a  in  (1)  will 
give 

{dx^ -\- dy^Y  dx~  {dx- -\- dy"^)- dy- 

R- 


( dxd-y  —  dyd-x )  -  ( dxd~y  —  dyd'~x)- 

{dx-^dy^ydx^            {dx^  ^  dy^Y  dy"" 
R'  = —  + ; 

(dxd^y  —  dyd'^xY         {dxd'^y  —  dyd'^xY 
{dx-  -^dy^Y 


therefore  R^  = 


{dxd^y  —  dyd^x)- 


or  R  =  ± ,  (2) 

[dxd-y  —  dyd-x) 

which  is  the  general  expression  for  the  value  of  the  radius  of 
the  osculatory  circle. 

If  dx  be  constant,  d-x^O  and  (2)  becomes 

^^^(,.^  +  ,fr^^  (3) 

dxd-y 

which  is  the  expression  for  the  value  of  the  radius  of  the 
osculatory  circle  applied  to  curves  referred  to  rectangular 
coordinates  in  which  the  abscissa  is  supposed  to  vary  uni- 
formly. 

Hence,  in  order  to  find  the  radius  of  curvature  for  any 
particular  curve,  the  first  and  second  rates  of  its  equation 
must  be  taken  and  the  values  of  dx,  dy,  d^y  obtained  and  substi- 
tuted in  (3). 

If  2  represents  the  arc,  then  (dx'^ -\- dy-)^''- ^=^  ds ;  substi- 
tuted in  (3),  this  gives 

d2^ 

R=± .  (4) 

dxd^y 

d-y 

It  has  been  shown  in  Art.  37  that  y  and ,  consequently 

dx- 

y  and  d-y,  have  contrary  signs  when  the  curve  is  concave 
toward  the  axis  of  abscissas  and  like  signs  when  convex; 
therefore,  if  we  wish  the  radius  of  curvature  and  the  ordinate 
of  the  curve  to  have  like  signs,  we  must  employ  the  minus 


ON  VARIABLE  QUANTITIES 


59 


sign  m  (2),  (3),  and  (4)  when  the  curve  is  concave  toward 
the  axis  of  abscissas  and  the  plus  sign  when  convex. 

If  P  and  P'  be  any  two  points  in  the  given  curve  APP'B, 

r  the  radius  of  the 
osculatory  circle  of 
the  point  P,  and  r'  the 
B  radius  of  the  point 
P'  (see  figure)  then 
curvature  at  P  :  curv- 

1     1 
ature    at    P' ::  — :  — : 
r     r' 

that  is,  the  curvature 
at  different  points  of 
a  curve  varies  in- 
versely as  the  radii  of 
the  osculatory  circles. 
53.  The  general  equation  of  Hues  of  the  second  order  is 
3/2  =^  ax-  -\-hx  -\-  c  ( 1 ) 

{2ax  -f-  &)  dx 


Fig.  ZQ 


and  its  rate 
therefore 

dx-  -j-  dy-  =  dx'- 


dy 


ly 


(2) 


{lax^hydx-       {\f- ^  {lax  ^  hy-\  dx"- 


or  {dx'' -\- dy-yi'' 


Ay-  4y2 

[43,2  _|_  {2ax  ^  by- Y'-' dx^ 


8y2 


(3) 


The  rate  of  (2)  is 

2aydx^  —  {2ax  -\-  h)  dxdy 


d-y  ^ 


2y^ 


{2ax  -f  b)  dx"" 
whence,  since  dxdy  ^ ;; [see  (2)],  by  substitut- 


ing it  and  reducing, 


^y 


[Aay""  —  (2ax -\- by]  dx^ 


43,3 

but  [see  (1)] 

4a3;2  =  4a^x^  +  4abx  +  4ac  =  {2ax  ^  b)-  +  Aac  —  b- 
or  403;^  —  {2ax  -\-  b)^  ^  Aac  —  b^ ; 


(4) 


60  AN  ELEMENTARY  TREATISE 

therefore,  substituting  this  in  (4)  and  multiplying  by  dx, 

(4ac  —  b-)  dx^ 
dxd'-y^ J^;^ •  (5) 

Substituting  (3)  and  (5)  in  (3)  of  Art.  52,  then 

[4y  +  {lax^hyyi- 

R  =  ± , 

2{4ac  —  h'') 

or,  substituting  for  y  its  value, 

{4{ax--^hx^c)-^{2ax-^hyyi- 

R  =  ±: ,  (6) 

2  ^4ac  —  b') 

which  is  the  general  expression  for  the  radius  of  curvature  of 
lines  of  the  second  order  for  any  abscissa  x. 

If  both  numerator  and  denominator  of  (6)  be  divided  by 
8,  then 

1 

[ax^  -\-bx  +  c  -{-—  {2ax  +  byy^^ 

R  =  ± -^ .  (7) 

ac  —  —  b" 
4 

The  numerator  of  this  value  of  R  is  the  cube  of  the  normal 
[see  (6),  Art.  40]  ;  therefore,  since  the  denominator  is  con- 
stant, it  is  evident  from  Art.  52  that  the  radii  of  curvature  at 
different  points  of  lines  of  the  second  order  are  to  each  other  as 
the  cubes  of  the  corresponding  normals. 

If  the  origin  of  coordinates  is  at  the  vertex  of  the  trans- 
verse axis,  c  =  0;  consequently,  using  the  minus  sign,  (6) 
becomes 

(4  {ax^  +  bx)  +  {2ax  +  byy^^ 

R  = , 

2b'- 

which,  when  jr  =  0,  reduces  to 

1 
R  =  —b. 

2 

In  this  case  b  is  the  parameter  of  the  curve;  therefore  the 
radius  of  curvature  at  the  vertex  of  the  transverse  axis  of  lines 
of  the  second  orde?  is  equal  to  half  the  parameter  of  the  curve. 

In  the  case  of  the  parabola  whose  equation  is 

y2  =  2pX, 


ON  VARIABLE  QUANTITIES  61 

a  =  0,  c  =  0,  b^2p;  therefore,  substituting  these  values  in 
(7)  and  using  the  minus  sign, 

R=z , 

which  is  the  general  value  of  the  radius  of  curvature  for  any 
point  of  the  parabola.  If  x  =  0,  then  R^=p,  the  radius  of 
curvature  at  the  vertex  of  the  axis. 

In  the  case  of  the  ellipse  whose  equation  is 

B^ 

,  &  =  0,  and  c  =  B- ;  therefore,  substituting  these 


A' 
values  in  (6),  reducing  and  using  the  minus  sign, 

(^4  _^2_^2_|_  52^2)3/2 


R 


A'B 


which  is  the  general  value  of  the  radius  of  curvature  for  any 
point  of  the  ellipse.     If  x  =  0,  then  R= ,  which  is  the 

radius  of  curvature  at  the  vertex  of  the  minor  axis,    li  x  =  A, 

B^ 
then  R  = ,  which  is  the  radius  of  curvature  at  the  vertex 

A 

of  the  major  axis. 

Taking  the  equation  of  the  logarithmic  curve, 
X  =  log  y, 
and  passing  to  the  rate  twice, 

dy  ydx 

dx=^M or  dy 


and  d^y  = 


y  M 

dxdy 


M 


dy^ 
whence  {dx^  +  dy^y-'^  = (M^  +  y^Y'K 

Substituting   the   values   of   dxd'-y   and    {dx- -\- dy-y-    in 


62  AN  ELEMENTARY  TREATISE 

(3)  of  Art.  52  and  using  the  plus  sign,  for  any  point  of  the 
logarithmic  curve 

R  ^= . 

My 

When  3;  is  equal  to  the  modulus  of  the  system  of  logarithms 
employed, 

R  =  2M  V2. 
From  the  ratal  equation  of  the  cycloid  (Art.  42) 

ydy 
dx  = ^-^ .  (1) 

{2ry  —  y-y^ 

Passing  to  the  rate  and  reducing, 

ydy'^  {r  —  3;) 
0=  {yd^y  +  dy^)  {2ry — y'^Y' 


{2ry  —  y~y'' 

{2ry  —  3;- )  yd-y  +  ryrfy^  ; 
whence 

rdy^  rydy^ 

d^y  =  — or  dxd~y  = 


2ry  —  y~  ( 2ry  —  y- )  ^'^ 

It  will  also  be  found  that 

y^dy^ 

dx^-  +  dy'  = +  dy^  = 

2ry  —  y 
y^dy^  -\-  2rydy^  —  y^dy^         2rydy^ 


therefore 


2ry  —  y^  2ry  —  y^ 

2rydy^  \/(2ry) 


{dx^ -\- dy^y^ 


(2ry  — y2)^/2 

Substituting  the  values  of  dxd^y  and  {dx"  +  dy^y^  in  (3) 
of  Art.  52  and  using  the  minus  sign  will  give 

R  =  2^{2ry); 

but  the  normal  is  equal  to  \/ {2ry)  by  Art.  43 ;  hence  the  radius 
of  curvature  for  any  point  of  the  cycloid  is  equal  to  twice  the 
normal  at  the  point  of  contact. 

EVOLUTES  AND   INVOLUTES 

54.  An  evolute  is  a  curve  from  which  a  thread  is  supposed 
to  be  unwound  or  evolved,  its  extremity  describing  another 
curve  called  an  involute. 


ON  VARIABLE  QUANTITIES 


63 


Thus,  let  a  thread  be  wrapped  about  the  curve  BCC'D  (Fig. 

21);  then,   if  the  thread  be  kept  tight  and  unwound   from 

p  BCC'D,   its   extremity,    com- 

mencing   at   A,   will   describe 

„    ^  >        5  the  curve  APP'S.    The  curve 

^^  \  BCC'D  is  called  the  evolute 

of  the  curve  APP'S  and 
APP'S  the  involute  of 
BCC'D. 

From  the  manner  in  which 


the  involute  is  generated  it  is 
evident  that  any  portion  of 
the  thread,  as  CP,  which  is 
disengaged  from  the  evolute 
is  a  tangent  to  it  at  C  and  per- 
pendicular to  the  involute  at 
P  ]  also,  that  any  point  in  the 
evolute,  as  C,  may  be  consid- 
ered as  a  center,  and  the  line 
CP  as  the  radius  of  a  circle  of  whose  circumference  that  por- 
tion of  the  involute  curve  at  P  is  an  arc. 

The  points  B,  C,  C  are  therefore  centers,  and  the  lines  BA, 
CP,  CP'  the  radii  of  circles  of  curvature  of  the  points  A,  P, 
P'  of  the  involute ;  hence  any  radius  of  curvature,  as  CP,  is 
equal  to  AB  plus  the  arc  BC  of  the  evolute. 

The  value  of  AB  will  depend  upon  the  position  of  the 
point  B,  from  which  the  arc  of  the  evolute  is  estimated;  but 
since  AB  is  the  radius  of  curvature  of  the  involute  at  A,  if  A 
is  the  origin  of  the  involute  and  B  the  corresponding  origin 
of  the  evolute,  B  will  be  the  center  of  the  osculatory  circle  to 
the  involute  at  its  origin.  Therefore,  if  the  radius  of  curva- 
ture at  the  origin  of  the  involute  is  equal  to  zero,  A  and  B  will 
coincide,  and  consequently  AB  will  be  equal  to  zero.  If  the 
involute  is  a  curve  of  the  second  order,  the  radius  of  curvature 
at  the  vertex  of  the  transverse  axis  is  equal  to  half  its  para- 
meter, Yzb,  by  Art.  53;  consequently  AB  will  be  equal  to  3'^&, 
and  B,  the  origin  of  the  evolute,  will  be  in  the  axis  of  abscissas 
AX. 

Hence,  since  the  center  of  any  circle  of  curvature  of  the 
curve  APP'S  is  in  the  curve  BCC'\D,  it  follows  that  the  equa- 
tion representing  the  coordinates  of  the  center  of  any  circle  of 
curvature  of  the  involute  will  be  the  equation  of  the  evolute. 


64  AN  ELEMENTARY  TREATISE 

Now  the  general  equation  of  the  circle,  consequently  of  any 
circle  of  curvature,  is 

{y  —  br  =  R'—{x  —  ay,  (1) 

in  which  a  and  b  are  the  coordinates  of  its  center  and  x  and  y 
the  coordinates  of  any  points  of  its  circumference;  therefore 
a  and  b  will  represent  the  coordinates  of  any  point,  as  C,  of  the 
evolute  BCC'D,  and  its  equation  will  be  6  =  /  (a)  ;  also  x  and 
y  will  represent  the  coordinates  of  any  corresponding  point,  as 
P,  of  the  involute  APP'S,  and  its  equation  will  be  3^  =  /  {x). 

Taking  the  rate  of  ( 1 )  twice 

(3'  —  b)  dy  =  —  {x  —  a)  dx 
and  ^3'"  +  (3'  —  b)d~y  =  —  dx- ; 

dx^  -\-  dy'^ 
whence  ^^3'  + (2) 

d'^y 

dy      dx^  -\-  dy- 

and  a  =  x  —  -— ( ).  (3) 

dx  d-y 

These  are  expressions  for  the  values  of  the  coordinates  of  the 
evolute  in  terms  of  the  rates  of  the  coordinates  of  the  involute. 

Hence,  if  we  take  the  rate  of  the  equation  of  the  involute 

dy 

twice,  y  =  f  {x),  obtain  the  values  of  dx",  dy-, ,  and  d-y, 

dx 

and  then  substitute  them  in  (2)  and  (3),  we  shall  have  two  new 
equations,  expressing  the  values  of  a  and  b,  the  coordinates  of 
the  evolute,  in  terms  of  x  and  y,  the  coordinates  of  the  involute. 

Finally,  by  combining  the  equations  thus  found  with  the 
equation  of  the  involute  and  eliminating  x  and  y,  an  equation 
will  be  obtained  containing  only  a  and  b,  which  will  be  the 
equation  of  the  evolute. 

Taking  the  equation  of  the  common  parabola 

y~  =  2px, 

and  passing  to  the  rate  twice 

ydy  =  pdx 

and  dy"^  -{-  yd-y  =  0 ; 

whence 

p-                       dy       p                              p- 
dx^-\-dy^=  ( +  1)  dx-, =  —,  2.ndd-y  =  — dx\ 

y^  dx        y  y^ 


ON  VARIABLE  QUANTITIES 


65 


Substituting  these  values   in    (2)    and    (3)    and   reducing 


give 


that 
and 


yO  yh  yi 

6  =  — or  6^  = ,  and  a^^^x  -^ -|-  p. 

f  P'  P 

Substituting  2px  for  y~  in  the  last  two  equations,  it  is  found 

—,  (4) 


6-  = 


1 


a=^^x  -\-  p  or  x=^ — (a  —  p). 


Finally,  substituting  —  (a  —  p)  for  x  in  (4),  then 


,-  =  —^a-py. 


(5) 


which  is  the  equation  of  the  evolute  and  shows  it  to  be  the 
semi-cubical  parabola. 

If  we  make  &  =  0, 
then  a  =  p;  hence,  the 
evolute  meets  the  axis  of 
abscissas  at  a  distance 
AB  from  the  origin 
(Fig.  22)  equal  to  half 
the  parameter  of  the  in- 
volute. 

If  the  origin  of  the 
coordinates  of  the  evo- 
lute be  transferred  from 
A  to  B,  (5)  becomes 


27p 

Since  every  value  of  a  gives  two  equal  values  of  b  with 
contrary  signs,  the  curve  is  symmetrical  with  respect  to  the 
axis  of  abscissas ;  the  evolute  BD^  corresponding  to  the  part 
AP  of  the  involute  and  BD  to  the  part  AP\ 

From  the  equations  relative  to  the  cycloid.  Art.  53,  it  is 
found  that 

2rydy^         dy         (2ry  —  3;^)  ^^ 
dx^  -f-  dy^ 


2ry 


dx 


y 


66 


AN  ELEMENTARY  TREATISE 


and 


d-y 


rd'f- 


2ry  —  3;2 

Substituting  these  values  in  (2)  and  (3)  of  Art.  54  will 
give 

h^=  —  y  and  a  =  x  -\-  2  (2ry  —  y^)"^^, 

whence  y  =  —  b  and  x  =  a  —  2  ( —  2rb  —  b^)^^^. 

Substituting*  these  values  of  x  and  3'  in  the  transcendental 
equation  of  the  cycloid  (Art.  42)  gives 

a  — 2  (— 2rb  —  b'y^  =  vers -^  (— b)  —  (—2rb  —  b-'y\ 

or  a  =  vers  -'  {— b)  +  (—  2rb  —  b- ) ^/% 

which   is   the   transcendental   equation   of   the   evolute   of   the 
cycloid,  referred  to  the  primitive  axes  and  origin. 

From  the  equation  of  the  radius  of  curvature  for  the  cy- 
cloid, R^2\/(2ry)  (see  Art.  53),  we  have  R  =  0  when 
3;:^0,  and  when  yr:=.2r  =  BD,R=^Ar-^A'B;  therefore  the 
origin  of  the  evolute  is  at  A,  and  A'D  =  BD. 

By  transferring  the  origin  of  the  coordinates  of  the  evolute 

from  A  to  A'  and 
Q  estimating   the   ab- 

scissas from  the 
right  toward  the 
left,  a  new  equation 
of  the  evolute  is 
formed  which  will 
be  found  to  be  of 
the  same  form  and 
to  involve  the  same 
constants  as  the 
equation  of  the  cy- 
cloid ;  hence  the 
evolute  of  a  cycloid  is  an  equal  cycloid — that  is,  the  arc  AA' 
is  a  facsimile  of  the  arc  AB,  and  A'C  of  the  arc  CB. 

Since  the  origin  of  the  evolute  is  at  A  and  the  radius  of 
curvature  for  the  vertex  B  of  the  cycloid  is  4r,  the  length  of 
the  evolute  AA'  is  4r;  hence  the  length  of  the  cycloid  ABC  is 
equal  to  8r,  or  four  times  the  diameter  of  the  generating  circle. 

EXAMPLES 

Determine  the  length  of  the  radius  of  curvature  for  a  point 
in  a  parabola  whose  abscissa  is  four  inches  and  ordinate  six 
inches. 


ON  VARIABLE  QUANTITIES 


67 


Determine  the  length  of  the  radius  of  curvature  for  a  point 
in  an  ellipse,  whose  abscissa  is  16  inches,  measured  from  the 
center,  the  semi-axes  being  26  and  13  inches. 

Determine  the  equation  of  the  evolute  of  the  equilateral 
hyperbola,  its  equation  being  y^=^x'^  —  A^. 

Determine  the  evolute  of  the  spiral  whose  ratal  equation  is 
dr 


dv 


ar 


Fig.    £4 


Curves  Referred  to  Polar  Coordinates 
55.  If  the  right  line  PC  (Fig.  24)  revolves  uniformly 
around  the  point  P,  and  if  at  the  same  time  a  point  moves  from 
P  along  the  line  PC  at  such  a  rate  that  at  the  first  revolution 
of  PC  it  will  arrive  at  A,  at  the  second  at  B,  etc.,  the  curve 
described  by  the  point  will  be  a  spiral. 

The  point  P  about  which  the  right  line  revolves  is  called 
the  pole ;  the  point  which  moves  along  the  line  PC  and  de- 
scribes the  curve  is  called 
the  generating  point;  a 
straight  line  drawn  from 
the  point  P  or  eye  of  the 
spiral,  so  called,  to  any 
point  of  the  curve,  as  A^,  is 
called  the  radius  vector, 
and  each  portion  of  the 
spiral  described  by  the 
generating  point,  as  PDA, 
AEB,  is  called  a  spire. 

With  the  pole  as  a 
center  and  PA  (the  dis- 
tance which  the  generating 
point  moves  from  P  along  PC  during  the  first  revolution  of 
PC)  as  a  radius,  if  the  circle  AFG  be  described,  the  angular 
motion  of  PC  about  the  pole,  consequently  the  radius  vector,  as 
PN,  is  measured  by  an  arc  of  this  circle,  estimated  from  A. 

Now,  if  r  represents  the  radius  vector  and  v  the  measuring 
arc  estimated  from  A,  it  is  evident  that  r  is  a  function  of  v  and 
may  generally  be  represented  by  the  equation, 

r  =  a?7",  ( 1 ) 

in  which  a  and  n  are  constants.    The  value  of  n  depends  upon 
the  law  which  governs  the  motion  of  the  generating  point  along 


68  AN  ELEMENTARY  TREATISE 

the  radius  vector  and  the  value  of  a  upon  the  relation  existing 
between  a  given  value  of  r  and  the  corresponding  value  of  v. 

If  n  is  positive,  the  spiral  represented  by  (1)  commences 
at  the  pole,  for  when  v^O,  r^O.  If  w  is  negative,  the  equa- 
tion becomes 

r^av-"";  (2) 

consequently  the  spiral  commences  at  an  infinite  distance  from 
the  pole,  for  when  v^O,  r  is  infinite,  or  when  r  =  0,  v  is 
infinite. 

When  n  is  equal  to  unity,  ( 1 )  becomes 

r^av.  (3) 

Now  if  a  =  AP,  the  circumference  of  the  circle  AFG  will 
be  2a  TT,  which  is  the  measuring  arc  for  the  first  revolution  of 
PC ;  therefore,  since  PA  or  a  is  then  the  radius  vector, 

a=^a-  2a  TT 

1 


whence 


27r 
Substituting  this  value  of  a  in  (3)  gives 

V 

2  TT 

the  equation  of  the  spiral  of  Archimedes. 

When  n  is  equal  to  one  half,  ( 1 )  becomes 

r  =  av^'  or  r^  ==  a^v, 

which  is  the  equation  of  the  parabolic  spiral,  being  of  the  same 
form  as  that  of  the  parabola;  for  substituting  y  for  r,  \/{2p) 
for  a,  and  x  for  i'  gives 

y^  =  2px.  (4) 

With  2p  as  radius  draw  the  circle  ABC,  divide  its  cir- 
cumference into  any  number  of  equal  parts,  as  six,  and  draw 
through  its  center  P,  the  divisional  lines  DD' ,  EE' ,  FF'.  With 

-(2/)  +  !),  -(2/>  +  2),  -(2/>-f  3),  and  -(2/^  +  4),   (1, 

2,  3,  and  4  being  values  given  x  as  in  the  construction  of  the 
parabola)  as  radii,  draw  the  arcs  Aa,  AhC,  Ac,  and  Ad,  having 
their  centers  in  the  line  AG ;  then  with  Pa,  Ph,  Pc,  and  Pd,  as 
radii,  draw  the  arcs  aa' ,  hh\  cc',  and  dd',  and  the  curve  drawn 


ON  VARIABLE  QUANTITIES 


69 


Fig.    Z5 


from  P  through  a',  h\  c' ,  d'  will  be  the  required  spiral.  In 
proof,  it  will  be  seen  that  AP :  PC  is  equal  to  Pb^  or  Ph'^  (see 
Euclid,  proposition  35,  Book  III)  ;  hence,  since  AP  =  2p  and 
PC  =  x,ii  y  be  represented  by  Pb  =  Pb\  then 

y^  =  2px. 

Also,  when  x  =  0,  3*  ^^  0 ;  therefore  the  spiral  commences  at 
P,  its  pole. 

When  n  is  equal  to  —  1,  (1)  becomes 

a 
r  =  av~'^  or  r  =  — .  (5) 


The  curve  represented  by  this  equation  is  called  the  hyper- 
bolic spiral  on  account  of  its  analogy  to  that  of  the  hyperbola 
when  referred  to  its  center  and  asymptote. 

With  a  as  a  radius  draw  the  circle  ABC  and  divide  its  cir- 
C  vJ  cumference,  2a  tt,  into  any 

number  of  equal  parts,  as 
six;   then   giving  to   v  the 

Sfl  TT  4a  TT 

values  2a  TT,  ,  , 

3  3 

air,  etc.,  the  corresponding 

1 

values    of   r   will   be  , 

27r 


70  AN  ELEMENTARY  TREATISE 

3        3       1 
— — ,  ,  — ,  etc.     Let  Pa,  Pb,  Pc,  Pd,  etc.  represent  these 

b  TT  4  TT  IT 

values  of  r;  then  the  curve  drawn  through  a,  h,  c,  d,  etc.,  Mrill 
be  the  hyperboHc  spiral. 

Take  any  point  in  the  spiral,  as  G,  and  draw  GH  perpen- 
dicular to  PL;  then  PG^r  and  the  angle  GPH  =  v;  hence 

GH  =  r  sin  v, 

or  substituting  for  r  its  value  from  (5) 

a  sin  V 
GH^ .  (6) 

V 

Now  it  is  evident  that  the  smaller  the  value  of  v,  the  nearer 
will  V  and  sin  v  approach  equality  and  consequently  the  nearer 
will  GH  become  equal  to  a;  therefore,  if  CJ  be  drawn  parallel 
to  PL,  CJ  must  be  an  asymptote  to  the  spiral. 

The  equation  of  the  logarithmic  spiral,  so  called,  is 

alogr^v.  (7) 

This  spiral  may  be  constructed  as  follows.  With  unity  for 
radius  draw  the  circle  ABC;  then,  giving  to  r  the  values  1,  2, 
3,  etc.,  the  corresponding  values  of  v  will  be  0,  a  log  2,  a  log  3. 


Fig.  17 

Set  off  from  A  on  the  circumference  of  the  circle  these  values 
of  V,  A,  Ab,  Ac;  then  to  A  and  through  b,  c,  draw  PA,  PD, 
PE,  the  values  of  r,  and  the  curve  drawn  through  A,  D,  E, 
will  be  the  logarithmic  spiral. 

Since  the  relation  between  r  and  v  is  entirely  arbitrary, 

P 
r^        ,  (8) 

1  -|-  cos  V 

is  the  polar  equation  of  the  parabola,  the  pole  being  at  the 
focus. 


ON  VARIABLE  QUANTITIES 


71 


The  polar  equation  of  the  eUipse,  the  pole  being  at  one  of 
the  foci,  is 

P 

r  =  - .  (9) 

1  +  ^  cos  V 

The  polar  equation  of  the  hyperbola,  the  pole  being  at  one 
of  the  foci,  is 

r  =  ~ .  (10) 

1  -|-  ^  COS  V 

In  (8),  (9),  and  (10)  p  represents  half  the  parameter,  e 
the  eccentricity,  and  v  the  angle  which  the  radius  vector  makes 
with  the  axis  of  X. 

Tangents  and  Normals 

56.  The  suhtangent  to  a  spiral  is  a  line  drawn  from  the 
pole  perpendicular  to  the  radius  vector  and  limited  by  a  tangent 
drawn  through  the  extremity  of  the  radius  vector ;  the  tangent 
is  a  line  extending  from  the  point  of  tangency  to  the  sub- 
tangent;  the  subnormal  is  a  line  drawn  from  the  pole  to  the 
foot  of  the  normal ;  the  normal  is  a  line  drawn  perpendicular 
to  the  tangent  and  extending  from  the  point  of  tangency  to  the 
subtangent  extended. 

Let  r  be  any  radius  vector,  as  PN  (see  Fig.  28),  v  the 
measuring  arc  estimated  from  A,  and  z  the  corresponding  arc 


of  a  spiral  of  which  BNC  is  a  section.  Then,  if  dz  be  repre- 
sented by  NM,  a  tangent  to  BNC  at  A'',  and  PN  be  extended 
to  N,  so  that  the  angle  NN'M  will  be  a  right  angle,  it  is  evident 
that  NN'  will  represent  dr,  the  rate  at  which  the  radius  vector 
PN  is  increasing.    Now,  draw  NM'  parallel  to  N'M  and  MM' 


72  AN  ELEMENTARY  TREATISE 

parallel  to  NN' ;  also,  with  F  as  a  center  and  PN  as  a  radius, 
describe  the  circular  arc  A'NS.  Then  it  will  be  seen  that  NM' , 
tangent  to  A'NS  at  N,  will  represent  the  rate  at  which  the  arc 
A'N  is  increasing  at  A''.  Hence,  since  NM'  represents  the  rate 
at  which  the  arc  A'N  is  increasing,  it  is  obvious  that  RR' , 
tangent  to  the  measuring  circle  at  R,  represents  c?z^,  the  rate  at 
which  V,  the  measuring  arc  estimated  from  A,  is  increasing  to 
correspond  with  NM  or  dz. 

Therefore,  since  the  triangles  PRR'  and  PNM'  are  similar, 
PR  :  RR'  ::PN:  NM', 
or,  making  the  radius  of  the  measuring  circle  unity, 

1 :  dv  : :  r :  NM', 
whence  NM'  =  rdv.  ( 1 ) 

Again,  from  the  similar  triangles  MM'N  and  NPT, 
MM' :  NM'  ::NP:  PT,  or,  since  NP  =  r,  MM'  =  NN'  =  dr, 
and  from  ( 1 ) ,  NM'  =  rdv, 

dr :  rdv  wr:  PT, 

r^dv 

whence  PT  =^ ;  (2) 

dr 

but  PT  is  the  subtangent  of  the  spiral,  hence : 

The  length  of  the  subtangent  to  any  point  of  a  spiral  is 
equal  to  the  square  of  the  radius  vector  into  the  rate  of  the 
measuring  arc,  divided  by  the  rate  of  the  radius  vector. 

For  the  tangent  TN , 

TN-  =PN^  -f  PT'-— 

r^dv^ 

that  is,  TN^  ^r~  -\- 

dr^ 

or  TN  =  ^{  dr^  +  r^dv^ )  ^-.  (  3  ) 

dr 

Hence  the  length  of  the  tangent  to  any  point  of  a  spiral  is 
equal  to  the  square  root  of  the  sum-  of  the  squares  of  the  radius 
vector  and  subtangent. 

For  the  subnormal  PQ, 

PT:PN::PN:FQ— 

r^dv 

that  is,  -.r-.-.r:  PQ 

dr 

dr 
or  PQ=-—.  (4) 

dv 


ON  VARIABLE  QUANTITIES  73 

Hence  the  length  of  the  subnormal  to  any  point  of  a  spiral  is 
equal  to  the  rate  of  the  radius  vector  divided  by  the  rate  of  the 
measuring  arc. 

For  the  normal  QN, 

dr''' 

that  is,  QN^  =  r'-\- 

dv'^ 

dr^ 
or  QN={r^~^--yK  (5) 

dv^ 

Hence  the  iength  of  the  normal  to  any  point  of  a  spiral  is 
equal  to  the  square  root  of  the  sum  of  the  squares  of  the  radius 
vector  and  subnormal. 

The  tangent  of  the  angle  of  tangency  of  a  spiral,  PTN, 

r~dv 

since  PN  =  r  and  PT  =^ from  (2),  is 

dr 

PN       dr 

= .  (6) 

PT       rdv 

Hence  the  tangent  of  the  angle  of  tangency  of  a  spiral  is 
equal  to  the  rate  of  the  radius  vector  divided  by  the  radius 
vector  into  the  rate  of  the  m^easuring  arc. 

PT 

The  tangent  of  the  angle  PNT  is  equal  to  :  but,  since 

b  fe  \i  pj^ 

r'-dv 


PN  =  r  and  PT 

dr 

PT       rdv 

PN        dr 
which  is  the  tangent  of  the  angle  the  tangent  line  makes  with 
the  radius  vector. 

Of  the  general  equation  of  spirals, 

r  =  av^, 

dr  dv  \ 

the  rate  is        — —  =  anv^~'^  or 


dv  dr 


74-  AN  ELEMENTARY  TREATISE 

dr  dv 

Substituting  the  value  of  or  ,  also  av^^  for  r,  in 

dv  dr 

formulas  (2),  (3),  (4),  (5),  (6),  and  (7),  the  result  will  be 


11  — 

n 

TN  = 

--{a 

2.^2)1      1 

av'' 
n 

{n- 

+  z;^)V3 

n     -\- 

2           ^        ~ 

PQ  =  anv"-^ 

QN  = 

{a- 

V'"  +  ( 

PN 

av^' 
n 

■1  (n 

2_^^2)y. 

PT 

av 

V 

PT       V 

PN        n 

V 

In  the  equation  of  the  spiral  of  Archimedes,  r  = ,  n=\, 

2  TT 

1 

and  a  = .     By  substituting  these  values  in  the  preceding 

2  TT 

formulas,  the  following  are  obtained : 

PT  =  ^,  TN  =  ^{\^v^-y\PQ=-^, 

Z  TT  Z  TT  Z  TT 

1  PN       I    PT 

QN  = (1  +  vy\ =  -, =  v. 

2m  ^      PT       V   PN 

If  v  =  2  7r — that  is,  if  the  tangent  is  drawn  at  the  extremity 
of  the  arc  generated  in  the  first  revolution  of  the  radius 
vector — then 

pr  =  2  TT— 

that  is,  PT  is  equal  to  the  circumference  of  the  measuring 
circle. 

At  the  completion  of  m  revolutions  v  =  2m^  tt,  and  conse- 
quently PT  =  2m^  TT  =  m  •  2m  TT — 
that  is,  at  the  completion  of  in  revolutions  the  subtangent  is 
equal  to  in  times  the  circumference  of  the  circle  described  with 
the  radius  vector  of  the  wth  revolution. 

In  the  equation  of  the  hyperbolic  spiral,  r^av,  n^  — 1  ; 
therefore  PT  =  —  a. 


ON  VARIABLE  QUANTITIES  75 

Hence  the  subtangent  of  the  hyperbolic  spiral  is  constant. 
From  the  equation  of  the  logarithmic  spiral, 

V  =  log  r, 

rdv 

it  will  be  found  that  =  M ; 

dr 

rdv 

but  [see  (7)]   represents  the  tangent  of  the  angle  made 

dr 

with  the  radius  vector  by  a  tangent  line  to  the  curve.  Hence 
the  tangent  of  the  angle  which  the  tangent  line  makes  with  the 
radius  vector  is  constant  and  equal  to  the  modulus  of  the 
system  of  logarithms  employed.  In  the  Naperian  system  the 
modulus  is  unity;  therefore,  if  v  is  the  Naperian  logarithm  of  r, 
the  angle  which  the  tangent  line  makes  with  the  radius  vector 
is  45°. 

Rate  of  the  Arc  and  Area  of  Spirals 
57.    Let  BNC  in  Fig.  29  be  a  section  of  a  spiral,  P  the 
pole  and  TN  a  tangent  to  the  curve  at  N.   Draw  PN,  and  NM' 
at  right  angles  to  PN ;  also  extend  TN  to  M  and  draw  MM' 
so  that  NM'M  will  be  a  right  angle ;  then 

NM^  =  M'M'  +  M'N^. 

But  since  NM  repre- 
sents ds;  M'M,  dr;  and 
M'N,  rdv  [see  (1)  of 
Art.  51], 

dz"=  dr^-{-  r^dv^  or  dz= 
(dr^  +  r'~dv'')\  (1) 

Hence  the  rate  of  an 
p  -T-     arc  of  a  spiral  is  equal 

to  the  square  root  of  the 
sum  of  the  squares  of  the  rate  of  the  radius  vector  and  of  the 
product  of  the  radius  vector  and  the  rate  of  the  measuring  arc. 
Let  PN  be  a  radius  vector  of  the  spiral  PBNC  in  Fig  30. 
Draw  NM'  and  M'P,  making  the  angle  PNM'  a  right  angle. 
Then,  representing  the  area  by  A, 

1 
dA=  —  PN-NM' 
2 

or,  since  PN  =  r  and,  by  (1)  of  Art.  51,  NM'  =  rdv, 

1 

dA=  —  r-dv.  (2) 

9  V   / 


76  AN  ELEMENTARY  TREATISE 


P    B 


This  is  evident  from  what  has-  been  shown  in  Art.  51 ;  for, 
since  NM'  represents  the  rate  at  which  the  arc  AN  is  increas- 
ing at  N,  it  must  also  represent  the  rate  at  which  the  extremity 
of  the  radius  vector  is  revolving  when  it  arrives  at  N.  Conse- 
quently the  area  of  the  triangle  PNM'  represents  the  rate  at 
which  the  area  of  the  spiral  is  increasing  when  the  radius 
vector  is  PN. 

Hence  the  rate  of  the  area  of  a  spiral  is  equal  to  one-half 
the  square  of  the  radius  vector  into  the  rate  of  the  measuring 
arc. 

EXAMPLES 

Determine  the  rate  of  an  arc  of  the  parabolic  spiral. 

Determine  the  rate  of  the  area  of  the  hyperbolic  spiral. 

If  the  rate  of  the  measuring  circle  of  the  Naperian  log- 
arithmic spiral  is  three,  at  what  rate  is  the  area  of  the  spiral 
increasing  when  the  radius  vector  is  four? 

Radius  of  Curvature  for  Spirals 

58.     Of    the    spiral    PNS    in    Fig.    31,    the    subtangent 

r'^dv 

PT  = (see       (2)       of      Art.      56),      the      tangent 

dr 

NT  =  ^^  {dr^  ^r^dv'-y^    [see    (3)    of  Art.  56],  the  radius 

dr 

vector  NP  =  r,   and   CN  =  R,   the   radius   of   the   osculator}^ 
circle  AMN ,  NQ  being  normal  to  the  spiral. 

Join  CP  and  draw  DP  parallel  and  BP  perpendicular  to 
NQ  ;  then,  since  BN  =  DP, 

CP^~  =  CN^  +  NP^~  —  2CN  ■  DP 
or  CP^  =  R-  +  r^  —  2R-DP  ; 

but    DP  =  r  s'mPND,    or,    since    s'm  PND    is    also    equal    to 


ON  VARIABLE  QUANTITIES 


77 


M 


Fjo.  31 


PT 


rdv 


NT 


^'    DP 


C 


A. 


{dr-  -f-  r^dv'^Y' 

CP^  =  i?2  _^  r^ 


2Rr-dv 


— :    therefore 

Vo  ' 


( dr~  -\-  r^dv- )  '''^ 


Now    the    equation    of    the    spiral    is    r- 


av^. 


(1) 
whence 


V  =^ ;  hence  dz/  ■ 


)/n^f. 


,i/« 


7.1/« 


Substituting  this  value  of  (/I/  in  (1),  then 


{r~'"  +  n-a^'^'Y^ 


(2) 


78  AN  ELEMENTARY  TREATISE 

Passing  to  the  rate,  CP  and  R  being  constant  for  any  point 
of  the  circle  AMP,  and  reducing, 

R  = ,  (3) 

which  is  the  general  value  of  the  radius  of  curvature  for  all 
spirals  represented  by  the  equation  r  =  av'\  in  terms  of  the 
radius  vector. 

In  the  case  of  the  spiral  of  Archimedes,  w=l,  and   (3) 

(r- 4- a-) 3/2 

becomes  R  = . 

r-  4-  2a2 

For  the  logarithmic  spiral,  whose  equation  is  log  r  =  v, 

Mdr 

dv  = ; 

r 

therefore,  substituting  this  value  oi  dv  m  (2),  it  will  be  found 

2RMr 
that  CP^  =  i?2  _|_  ^2 


(M2  +  \y- 

Passing  to  the  rate  and  reducing, 

R  == . 

M 

If  the  Naperian  system  be  used,  M  =  1,  and  R  =  r  \/2. 

Determine  the  radius  of  curvature  for  a  parabolic  spiral; 
also  for  the  hyperbolic  spiral. 

Singular  Points  of  Curves 

59.  It  has  been  shown  in  Art.  36  that  the  first  ratal  co- 
efficient of  the  equation  of  a  curve  represents  the  tangent  of 
the  angle  of  tangency;  therefore,  since  the  tangent  of  this 
angle  is  zero  when  the  angle  is  zero,  and  infinite  when  the 
angle  is  90°,  it  follows  that  the  roots  of  the  equation 

dy 

dx 
will  give  the  abscissas  of  all  points  of  the  curve  at  which  the 


ON  VARIABLE  QUANTITIES 


79 


tangent  line  is  parallel  to  the  axis  of  abscissas;  also  that  the 
roots  of  the  equation 

dy 

dx 

will  give  the  abscissas  of  all  points  of  the  curve  at  which  the 
tangent  line  is  perpendicular  to  the  axis  of  abscissas  (see  Fig. 
32  and  Fig.  Z^). 

Taking  the  equation  of  the 
circle 

3;=±  {R'  —  x'^y- 
and  passing  to  the  rate, 
dy                        X 
-^■=-^ •  (1) 

If  (1)  =0,  ;r  =  0,  but  when 
x  =  0,  y=±R;  therefore  the 
circle  has  two  tangents  parallel 
to  the  axis  of  abscissas,  (see  Fig. 
32).  If  (1)  is  infinite,  ^  =  ±  i? 
and  3;  =  0 ;  therefore  the  circle 
has  two  tangents  perpendicular 
to  its  axis  of  abscissas  (see  Fig. 
33). 

Of    the    equation    of    the      Y 
parabola, 

y=±yy{2px), 
the  rate  is 

dy  p 

dx 


(2) 


V(2M) 
If  (2)=0,  both  .r  and  3; 
are    infinite;     therefore    the 
parabola     has     no     tangent  F"'g-  ^^ 

parallel    to    its    axis    of    ab- 
scissas.    If   (2)   is  infinite,  both  x  and  3;  are  equal  to  zero; 
therefore  the  parabola  has  a  tangent  (the  axis  of  ordinates), 
perpendicular   to    its    axis   of   abscissas   at   the   origin   of    its 
coordinates,  as  shown  in  Fig.  34. 

Points  of  Inflection 
60.   Those  points  of  a  curve  at  which  the  curve  changes  its 
direction — that  is,  from  being  concave  to  its  axis  of  abscissas  it 


80  AN  ELEMENTARY  TREATISE 

becomes  convex,  or  vice  versa— are  called  points  of  inflection. 
At  such  a  point  the  angle  of  tangency  and  consequently 
its  tangent  must  either  change  from  increasing  to  decreasing, 
or  from  decreasing  to  increasing;  therefore  the  rate  of  varia- 
tion of  the  tangent  of  the  angle  of  tangency  at  a  point  of 
inflection  will  be  zero,  real,  or  infinite;  zero  when  the  angle  of 
tangency  is  zero,  real  between  0°  and  90°,  and  infinite  when 

90°.    Hence  since  represents  the  rate  of  variation  of  the 

tangent  of  the  angle  of  tangency,  every  point  of  inflection  will 
have  for  its  abscissa  some  root  of  the  equations : 

-^^0  (1),  \0  (2),  and  =  oo    (3). 

dx'  dx-  dx^ 

But  it  does  not  follow  that  every  root  of  these  equations  will  be 

the  abscissa  of  a  point  of  inflection ;  hence  it  is  necessary  to 

d^y 

examine  whether  the  value  of  x  will  give contrary  signs 

dx'~ 

(see  Art.  37). 

Let  the  equation  of  the  curve  be 

y^a  +  b  (x  —  c)^  (4) 

Then  passing  to  the  rate  twice, 

dy 

-^=.Zb{x  —  cy  (5) 

dx 

d^y 

and  =^6b  {x — c).  (6) 

dx~ 

Making  (6)  equal  to  zero,  then 

but  when  x  =  c,  the  first  ratal  coefficient  is  equal  to  zero  also ; 
therefore  y  =  a  when  x^=^  c,  there  is  a  tangent  line  to  the 
curve  at  the  point  whose  coordinates  are  a  and  c,  which  is 
parallel  to  the  axis  of  abscissas. 


ON  VARIABLE  QUANTITIES 


81 


JJ 


If  h  is  positive,  the  second  ratal  coefficient  will  be  zero  for 
x  =  c,   but  negative   when  x  <^c   and  positive  when  x ^  c ; 

therefore  there  is  an  inflection 
of  the  curve  at  the  point  whose 
abscissa  is  ;ir^c  (see  Fig.  35). 
If  6  is  negative,  the  second 
ratal  coefficient  will  be  positive 
when  .r  <;;  c  and  negative  when 
X  y.  c ;  therefore,  at  the  point  of 
the  curve  whose  abscissa  is 
x^  c,  there  is  an  inflection  of 
the  curve,  but  opposite  to  the 
first  (see  Fig.  36). 
^  ^  In  the  first  case  the  curve  is 

first  concave,  then  convex  to  the 
axis  of  X;  in  the  second  case  it 
is  first  convex,  then  concave,  as 
A  X  shown  in  the  figures. 

Let  the  equation  of  the  curve  be 

y=^a-\-h  (x  —  c)^/^. 

Then,  passing  to  the  rate  twice. 


dy 


3b 


dx 


and 


d^y 


dx^ 


25  (x  —  c)' 


5  7 


5  {x  —  cy^' 

Making  x^  c,  both  expres- 
sions become  infinite ;  therefore 
the  first  ratal  coefficient,  since 
y^a  when  x^  c,  gives  a  tan- 
gent line  to  the  curve  at  the  point 
whose  coordinates  are  a  and  c, 
which  is  perpendicular  to  the 
axis  of  X. 

If  b  is  positive,  the  second 
ratal  coefficient  will  be  positive 
for  all  values  of  jt  <  c,  and  nega- 
tive for  all  values  of  x  ^  c; 
hence,  for  all  values  of  x  less 
than  c,  which  makes  y  positive, 
the  curve  will  be  convex  to  the 
axis  of  X,  while  for  all  values  of 
X  greater  than  c,  it  will  be  concave  (see  Fig.  37). 

If  b  is  negative,  the  case  will  be  the  reverse,  as  shown  in 
Fig.  38. 


82 


AN  ELEMENTARY  TREATISE 


If  a  =  0,  P  will  be  the  point  of  inflection,  and  if  o  =  0, 
also  c^=0,  A  will  be  the  point  of  inflection. 

Cusp  Points 

61.  The  point  at  which  two  branches  of  a  curve  terminate 
and  have  a  common  tangent  is  called  a  cusp  point.  When  the 
cusp  is  formed  by  the  union  of  two  branches,  one  on  either 
side  of  the  tangent,  it  is  called  a  cusp  of  the  first  order, 
and  when  both  branches  are  on  the  same  side  of  the  tangent, 
it  is  called  a  cusp  of  the  second  order. 

If  x  =  c  be  the  abscissa  of  a  cusp  point,  the  values  of  x 
immediately  preceding  and  following  that  of  x^  c,  when 
substituted  in  the  given  equation,  will  give  to  3*  either  two  real 
or  two  imaginary  values ;  if  real,  both  will  be  greater  or  both 
less  than  that  of  the  cusp  point;  furthermore,  for  a  cusp  point 
there  will  be  a  distinguishable  term  in  the  second  ratal  co- 
efficient, either  equal  to  zero  or  infinity. 

Let  the  equation  of  the  curve  be 

3;  =  a.ar  ±  h  {x  —  c)°'^ ; 

then,  taking  the  rate  twice  gives 

dy  5  d-y 

a  ±  —  h  {x  —  c)"'-  and 


15 
4 


h  {x  —  cy\ 


dx  2  dx'- 

Making  the  second  ratal  coefficient  equal  to  zero,  we  have 
x=  c;  hence,  since  for  a  value  of  x  less  than  c,  y  will  have  two 
imaginary  values,  and  for  a  value  of  x  greater  than  c,  y  will 
have  two  real  values,  there  is  a  cusp  at  the  point  of  the  curve 
whose  abscissa  is  x  =  c  (see  Fig.  39). 

When  x  =  c,  the  first  ratal  coefficient  equals  a ;  hence  the 
tangent  of  the  angle  of  tangency  at  the  cusp  point  is  equal  to  a, 

and  since  y  =  ac  when  x^=  c, 
the  tangent  line  to  the  curve  at 
the  cusp  passes  through  the 
origin  of  coordinates.  Also  for 
any  value  of  x  greater  than  c, 
d^y 

will  have  two  values,  one 

dx'' 

positive  and  the  other  negative ; 
consequently  one  branch  of  the 
curve  is  convex  and  the  other 
concave  to  the  axis  of  X ;  there- 
V  fore  a  branch  must  lie  on  either 
side  of  the  tangent  line  AN  and 
the  cusp  is  of  the  first  order. 


ON  VARIABLE  QUANTITIES 


83 


The  equation  of  the  semi-cubical  parabola  is 


y^±ax 


3/2 


the  rates  of  which  are 
dy 

dx 

d^y 


ax' 


and 


dx^ 


3a 

ArX'^' 


Making 


dx^ 


CO ,  jt;  =  0 ;  then  y  has  two  imaginary  values 


for  ;r<^0  and  two  real  values  for  x^O;  and,   since  ^  =  0 

when  x^O,  there  is  a  cusp  at  the  origin  of  the  coordinates; 

dy 

but  when  x^O, =  0,  hence  the  axis  of  abscissas  is  a  tan- 

dx 

gent  to  both  branches  of  the  curve  at  the  cusp  (see  Fig.  40). 
Examination  of  the  primitive  equation  shows  that  for  every 

value  of  X  greater  than  0,  3;  has  two  values,  one  positive  and 
the  other  negative ;  therefore  one 
branch  of  the  curve,  AC,  lies  above, 
and  the  other,  AC ,  below  the  axis  of 
X,  and  the  cusp  is  of  the  first  order. 

d^y 
vy  For  any  value  of  jt'^  0,  has  two 

values,  one  positive  and  the  other  neg- 
ative ;  consequently,  since  y  is  negative 
for  the  branch  AC ,  both  branches  are 
convex  to  the  axis  of  X. 


Fig.  4  0 


C 
Of  the  equation 

the  rates  are 


y=^a-\-  h  (x  —  c) ^^^, 
dy  2b 


and 


dx 
d^y 


3  (^x  —  cy^ 

2b 


dx^ 


9  {x  —  cy^ 

Making  the  second  ratal  coefficient  equal  infinity.  x  =  c ; 
hence,  since  y  =  a  when  x-^  c,  and  since  y  is  greater  than  a 
either  for  .r  <^  c  or  x  "^  c,  there  is  a  cusp  at  the  point  of  the 
curve  whose  coordinates  are  x  =  c,  y^a. 

When  the  first  ratal  coefficient  is  equal  to  infinity,  jr  =  c ; 
hence  a  tangent  line  to  the  curve  at  the  cusp  point  is  perpen- 
dicular to  the  axis  of  abscissas  (see  Fig.  41). 


AN  ELEMENTARY  TREATISE 


Of  the  equation 


the  rates  are 


and 


For    any  value   of   x,   either 
less  or  greater  than  c,  the  value 

of is  negative;  consequently 

both  branches  of  the  curve  are 
concave  to  the  axis  of  abscissas ; 
also,  since  y  has  a  value  corre- 
sponding to  either  x  <^c  or 
X  ■^  c,  a  branch  lies  on  either 
side  of  the  tangent  line  TG. 

d^y 

If  b  is  negative,  then be- 

dx^ 

comes  positive  for  any  value  of 

X,  either  less  or  greater  than  c ; 

therefore   both   branches   of   the 

curve  are  convex  to  the  axis  of 

abscissas  (see  Fig.  42). 


4 


^  5 


dy 


dx 

d'y 


Ax 


9-V-3/2 


dx' 


4  ±  Zx'^^ 


Making  the  first  ratal  coefficient  equal  to  zero,  then  ^  =  0 

32        288 
or  4 :  but  when  .r  ^  0,  -y  ^  0,  and  when  .r  =  4,  3/  =  — -  or  — — ; 

5  5 

therefore  the  axis  of  abscissas  is  tangent  to  the  curve  at  A, 

the  origin.     There  is  another 

tangent    to    the    curve    at    E, 

parallel  to  the  axis  of  X,  and 

corresponding  to  an  abscissa 

32 
of  4  and  an  ordinate  of  — — . 

0 

If    the    positive    sign    be 

d^y 

used   (since  will  then  be 

dx- 


ON  VARIABLE  QUANTITIES  85 

positive  for  any  value  of  x),  the  left  hand  branch  of  the  curve, 
AB,  is  convex  to  the  axis  of  abscissas.  But  if  the  negative 
sign  be  used  which  corresponds  with  the  right  hand  branch  of 

d-y  16 

the  curve  AEC,  since  will  then  be  positive  for  x  <'~Z~ 

dx'  9 

16 
and  negative  for  x^ ,  the  part  AD,  answering  to  ;ir  =  U  to 

16 

,  will  be  convex  and  the  part  DEC  concave;  consequently 

9 

there  is  an  inflection  at  D. 

In  conclusion,  since  3;  has  two  imaginary  values  for  ^  <^0, 
and  two  real  values  for  x  '^0,  and  since  y  =  0  when  x^O, 
the  branches  AB  and  AEC  form  a  cusp  of  the  second  order  at 
A,  their  origin. 

Multiple  Points 

62.  The  points  at  which  two  or  more  branches  of  a  curve 
intersect  are  called  a  multiple  point. 

At  a  multiple  point  it  is  therefore  evident  that  there  must 
be  as  many  tangents  to  the  curve  as  there  are  intersecting 

dy 
branches ;  hence ,  which  represents  the  tangent  of  the  angle 

dx 
of  tangency,  will  have  as  many  values  as  there  are  different 

tangents. 

Let  the  equation  of  the  curve  be 

y  =  a±x  {h^  —  x'-y^;  (1) 

then,  passing  to  the  rate, 

dy  b^  —  2x~ 

-^=± .  (2) 

dx  {b~  —  x^)'''^ 

An  inspection  of  ( 1 )  shows  that  values  of  x  greater  than  b 
make  y  imaginary,  while  for  values  of  x  less  than  b,  y  has  two 
values ;  hence  the  curve  has  two  branches  which,  since  y=^a 


86 


AN  ELEMENTARY  TREATISE 


when  x  =  0,  intersect  the  axis  of  ordinates  at  a  distance  a  from 
A,  the  origin  (see  Fig.  44). 


dy 

When  x=  b  or  —  b, becomes  infinite,  consequently  the 

dx 

tangents  to  the  curve  at  B  and  B^  are  perpendicular  to  the  axis 

dy 

of  abscissas.     When  x^O,  ^±  b;    therefore  there  are 

dx 

two  tangents,  TN  and  T'N',  to  the  curve  passing  through  the 

multiple  point. 


Let  the  equation  of  the  curve  be 

3^=1+  (IzpV-^)  (i±V^)^s 
then,  passing  to  the  rate, 

dy  I  zf.S\/  X 


dx 


4  (y/  X  d=  xy 


(3) 


(4) 


From  an  examination  of  (3)  we  find  for  .*:  .^0  that  3^  is 
imaginary ;  f or  jr  ^  0,  y  ^  0  or  2 ;  for  any  value  between  0  and 
1,  that  y  has  four  real  values ;  for  x==l,  3*  =  1 ;  and  for  any 
value  of  X  greater  than  1,  that  y  has  only  two  real  values. 


ON  VARIABLE  QUANTITIES 


87 


Hence  two  branches  of  the  curve  must  intersect  each  other  at 
the  point  whose  coordinates  are  x=^\  and  y=\  (see  Fig.  45) . 

dy  1  1 

When  x=l, is  infinite,  +  — \/2,  or  —  —  \/2,  conse- 

dx  2  2 

quently  the  curve  has  a  multiple  point  corresponding  to  the 
coordinates  x=l,  y=l,  and  at  this  point  there  are  three 
Y 


tangent  lines,  TG,  T'G' ,  T"G" :  TG  is  perpendicular  to  the 
axis  of  abscissas,  and  T'G'  and  VG"  make  angles  therewith, 

1  1 

whose  respective  tangents  are  -f-  —  \/2  and  —  —  \/2. 

Isolated  Points 
63.    A  point  which  is  entirely  detached  from  a  curve,  but 
whose  coordinates  satisfy  the  equation,  is  called  an  isolated  or 
conjugate  point. 

Since  a  point  entirely  detached  from  a  curve  can  have  no 
tangent,  it  is  evident  that  for  an  isolated  point,  the  first  ratal 
coefficient  of  the  equation  will  be  imaginary. 

Let  the  equation  be 

y=.±{x^a)^x;  (1) 

then,  passing  to  the  rate, 

dy  2)X  -\-  a 

(2) 


dx 


2^x 


AN  ELEMENTARY  TREATISE 


By  examining  (1)  we  find  that  x  =  0  makes  3'  =  0,  and 
for  any  value  of  jr  >  0,  y  has  two 
real  values,  one  positive  and  the  other 
negative;  therefore  the  curve  passes 
through  the  origin  A  and  has  two 
branches,  AC,  AC,  extending  to  the 
right  (see  Fig.  46). 

Equation    (1)    is  also  satisfied  by 
the  coordinates  x^  —  a  and  3'^0; 

dy 

but  when  ;r  =  —  a,  becomes  im- 

dx 

aginary;  hence,  the  point  P,  whose 
abscissa  is  x  =  —  a,  being  entirely 
detached  from  the  curve,  is  an  iso- 
lated point. 

The  rate  of  (2)  is 

d~y  3  X  —  a 


dx- 


^x^x 


Making  this  equal  to  zero  gives  x=^  —  a\  therefore  there  is 

1 
an  inflection  of  the  curve  at  the  point  whose  abscissa  is  .r  =  —  a. 


Maxima  and  Minima 

64.  If  a  variable  quantity  increases  until  it  attains  a  value 
greater  than  any  immediately  preceding  or  following  it,  such 
a  value  is  called  a  maximum ;  and  if  it  decreases  until  it  attains 

a  value  less  than  any  imme- 
diately preceding  or  following 
it,  such  a  value  is  called  a 
minimum. 

Illustration.  Let  the  points 
P  and  P'  be  so  situated  in  the 
curve  BPP'C  (see  Fig.  47) 
^  that  tangent  lines,  T'N  and 
T'l^' ,  to  the  curve  at  P  and 
P' ,  shall  be  parallel  to  the  axis 
of  abscissas,  AX;  then,  since  it  is  obvious  that  the  ordinate 
FP  is  greater  than  any  immediately  preceding  or  following  it, 
and  that  the  ordinate  F'P'  is  less  than  any  immediately  pre- 
ceding or  following  it,  FP  is  a  maximum  and  F'P'  a  minimum. 


A.B 


ON  VARIABLE  QUANTITIES  89 

Therefore,  if  y  =  /  {x)  is  the  equation  of  the  curve,  x  repre- 
senting any  abscissa  and  y  the  corresponding  ordinate,  y  is  a 
maximum  when  it  is  equal  to  F'P\ 

Now,  for  that  point  of  a  curve  at  which  a  tangent  Hne  is 
parallel  to  the  axis  of  abscissas,  since  it  then  makes  no  angle 
with  this  axis,  and  since  the  first  ratal  coefficient  of  its  equa- 
tion represents  the  tangent  of  the  angle  of  tangency  (see  Art. 
36),      • 

dy 

dx 

Hence  when  a  function  is  either  a  maximum  or  a  minimum 
the  first  ratal  coefficient  is  equal  to  zero.  For  instance,  if  the 
function  is  of  the  form 

y  =  x^  —  2ax  -\-  b, 
the  rate  is 

dy 

=  2x  —  2a ; 

dx 

therefore  y  is  either  a  maximum  or  a  minimum  when 

2x  —  2a  ^  0  or  when  x=^a. 

It  will  be  observed  that  the  curve  BPP'C  is  concave  to  its 
axis  of  abscissas  at  the  point  P,  and  convex  at  the  point  P' ; 
therefore,  from  Art.  37,  since  either  ordinate  FP  or  F'P'  is 
positive,  the  second  ratal  coefficient  of  the  equation  of  the 
curve  will  be  negative  for  the  ordinate  FP  and  positive  for  the 
ordinate  F'P\  But  it  has  been  shown  that  FP  is  a  maximum 
ordinate,  and  F'P'  a  minimum  ordinate;  consequently,  if  the 

d^y 

equation  of  the  curve  is  y^f  {x), will  be  negative  when 

dx^ 

3/  is  a  maximum  and  positive  when  3;  is  a  minimum. 

Hence,  to  find  the  values  of  the  variable  of  a  function  which 
will  render  the  function  a  maximum  or  a  minimum,  also  to 
distinguish  the  one  from  the  other,  we  have  the  following  rule. 

Make  the  first  ratal  coefficient  of  the  function  equal  to 
zero  and  find  the  values  of  the  variable  in  this  equation;  then 
substitute  these  values  in  the  second  ratal  coefficient  of  the 
function,  and  each  value  which  gives  a  negative  result  will 
render  the  function  a  maximum,  and  each  value  which  gives 
a  positive  result  will  render  it  a  minimum. 


90  AN  ELEMENTARY  TREATISE 

It  sometimes  happens  that  a  value  of  the  variable,  as  jt  ^  a, 
found  by  making  the  first  ratal  coefficient  of  the  function  equal 
to  zero,  will  reduce  the  second  ratal  coefficient  to  zero  also.  In 

this  case,  substitute  a  ±v  for  x  in  ,  and  if  either  -\-  v  ov 

dx'- 

—  V  give  a  negative  result  for  a  small  value  of  v,  y  will  be  a 
maximum;  but  if  the  result  is  positive,  3;  will  be  a  minimum. 
If  one  sign  gives  a  negative  result  and  the  other  a  positive 
result,  it  is  clear  y  will  be  neither  a  maximum  nor  a  minimum ; 
such  a  result  simply  indicates  that  the  curve  represented  by  the 
proposed  equation,  has  an  inflection  at  the  point  corresponding 
to  the  abscissa  x^a  (see  Art.  60). 

For  illustration,  take  the  equation 

y  =  x'  —  Ax""  +  \6x  +  13 ; 
then,  passing  to  the  rate  twice, 
dy 


dx 


4x'  —  l2x'  +  l6,  (1) 


and  -^=12.^2  —  24;^.  (2) 

dx-' 

If  (1)=0,  then  x  =  —l  or  .r  =  2. 

Substituting  these  values  of  x  in   (2),  then  for  x^ — 1, 

d^y  d^y 

36  and  for  x  =  2,  =  0. 


dx^  dx' 

Since  x^2  reduces  the  second  ratal  coefficient  to  zero,  by 
substituting  2±v  for  x  in  (2)  the  result  is  \2v{v±2), 
which,  for  a  small  value  of  v,  is  negative  for  the  minus  sign 
and  positive  for  the  plus  sign,  which  shows  that  there  is  an 
inflection  of  the  curve  at  r' ,  corresponding  to  the  abscissa 
x  =  2;  hence  this  value  of  x  makes  y  neither  a  maximum  nor 
a  minimum  (see  Fig.  48).  There  is  also  an  inflection  at  r,  but 
a  minimum  for  3^  answering  to  .r^ —  1. 


ON  VARIABLE  QUANTITIES 

Y 

V\a.  48 


91 


Let  the  equation  be 

i(3^2_i2) 

ay         I 

then       = ^  0,  whence  x  =^  ±2. 

dx  {^x^ — \2xy'~ 

Taking     the     rate     of      (3)      and     reducing,     regarding 
3jir^  —  12^0,  the  result  is 

dry  Zx 

dx''-        (jr"  —  12:1;)'''^' 

3 

which,    for    ;»;  =  — 2,    equals    — — ,    and    for    ;f  =  2,    equals 

3  .  ^ 

—  —  V —  1 ;  therefore  a;  is  a  maximum  when  x^=  —  2  and  a 

2 

minimum  when  x-^  -\-2. 

If  3;2___^___^3 \2x, 

dv 


then 


dx 


3x'-  —  l2; 


whence  x  ^  ±:  2, 

the  same  values  of  x  as  before. 


92  AN  ELEMENTARY  TREATISE 

Hence  when  the  expression  containing  the  variable  is  under 
a  radical,  the  radical  may  be  omitted. 

65.   In  case  the  equation  is  of  the  form 

f{x,y)=0,  (1) 

in  which  3;  is  a  function  of  x,  pass  to  the  rate  and  find  from 

dy 
^  0  the  value  of  x  in  terms  of  y,  also  of  y  in  terms  of  x. 

dx 

Substitute  the  value  of  x  in  terms  of  3/  in  (1)  and  therefrom 
determine  the  value  of  y,  and  with  this  value  of  y  find  that  of  x. 
Next  determine  the  second  ratal  coefficient  and  substitute 
therein  the  values  of  x  and  y  found  from  (1).  And  if  the 
result  is  negative,  y  will  be  a  maximum ;  but  if  positive,  a  mini- 
mum. Should  these  values  reduce  the  second  ratal  coefficient 
to  zero,  proceed  as  in  article  64. 

EXAMPLE 

y'^-\-2axy  —  x^  —  6^  =  0.  (2) 

Passing  to  the  rate, 

2ydy  -j-  2axdy  -)-  2aydx  —  2xdx  =  0, 

dy         X  —  ay 

-j^  = —,  (3) 

dx         ax  -\-  y 

whence,  by  making  it  equal  to  zero, 

X 

x^ay  or  y^  — . 
a 

Substituting  ay  for  x  m.  (2),  we  find 

b 

y  = 


therefore  x  ■ 


ab 


dy 

Taking  the  rate  of  (3) ,  regarding =  0,  also  x  — ay  =  0, 

dx 

d^y  ax  -\-  y  1 

then 


dx^        {ax  -\-  yY         QX  -\-  y 
or,  substituting  for  x  and  3;  their  values  and  reducing, 

d'y  1 

dx^  ~  b  (1  +a2)V.  ■ 


ON  VARIABLE  QUANTITIES  93 

ah 


Hence  3;  is  a  minimum  when  x  = 


(l+a'-r 


dy 

The  value  of  ;ir  =  a  found  from  =  0  will  sometimes 

dx 

make  the  second  ratal  coefficient  infinite.    In  such  a  case,  sub- 

d^y 
stitute  a  ±  V  (v  being  a  small  quantity)  for  x  in .    Then 

dx^ 

if  the  result  for  both  signs  of  v  is  negative,  y  will  be  a  maxi- 
mum for  x  =  a,  and  if  positive,  a  minimum ;  but  if  the  result 
for  one  sign  is  negative  and  for  the  other  sign  positive,  3;  will 
be  neither  a  maximum  nor  a  minimum. 

Let  the  function  be 

y==b — (x  —  a)*/^. 

Passing  to  the  rate  twice, 

dy  4 

~-  =  —-{x  —  ayr^,  (4) 

dx  6 

d'~y  4 

and  for  x  =  a,        =  — =  co.  (5) 

dx'~  9  (x  —  ay/' 

If  (4)  ^0,  x^a;  but  when  x^=a,  (5)  becomes  infinite; 
therefore,  by  substituting  a  ±:  v  for  ;ir  in  (5), 

d^y  4 

dx^~        9(±vy/^' 

which  is  negative  for  either  plus  or  minus  v;  therefore  y  is  a 
maximum  when  x^a. 

66.  When  the  curve  represented  by  a  given  equation  forms  a 
cusp  of  the  first  order,  and  a  tangent  line  to  the  curve  at  the 
cusp  point  makes  an  angle  of  ninety  degrees  with  the  axis  of 
abscissas,  it  is  evident,  as  may  be  seen  in  the  figures  41  and  42, 
that  the  ordinate  of  the  cusp  point  will  be  a  maximum  or  mini- 
mum, according  as  both  branches  of  the  curve  are  concave  or 
convex  to  the  axis  of  abscissas ;  but  when  the  angle  of  tangency 
is  90°,  the  first  ratal  coefficient  of  the  equation  is  infinite. 
Therefore,  to  find  the  value  of  the  variable  which  will  render 
the  function  a  maximum  or  a  minimum  in  such  a  case,  a  solu- 
tion is  required  of 

dy 
=  00. 

dx 


94  AN  ELEMENTARY  TREATISE 

If  the  value  of  x  thus  found  renders  the  second  ratal  co- 
efficient infinite,  proceed  as  has  been  previously  explained. 

For  an  example,  take 

y  =  a  —  9  (x  —  cy/^; 
then,  passing  to  the  rate  twice, 

dy  6 


dx  (x  —  c) 


1/3 


(1) 


d^y  2 

and  = =  00.  (2) 

dx^       (x — c)*/^ 

( 1 )  is  satisfied  when  x^  c,  and  by  substituting  c  -]-  v  m 
(2),  it  becomes 

d'y  _    2 

dx-  ~  z/*/^  ' 

therefore  3;  is  a  minimum  when  x=^  c. 

EXAMPLES 

Determine  the  values  of  the  variable  that  will  make  the 
following  maxima  or  minima. 

1.  y  =  x^  —  8x''^22x''  —  24x. 

2.  y  =  b—(x  —  a)'/\ 

.3.  y  =  4  ±  (3x'- —  Ux -{- 9y^^ 

4.  Divide  a  quantity,  a,  into  two  such  parts,  that  the  wth 
power  of  one  part  multipHed  by  the  nih.  power  of  the  other 
part  shall  be  a  maximum. 

5.  Determine  the  minimum  hypotenuse  of  a  right-angled 
triangle  containing  an  inscribed  rectangle  whose  sides  are  as 
a  to  b. 

6.  Determine  the  length  of  the  axis  of  the  largest  parabola 
that  can  be  cut  from  a  right  cone,  the  length  of  whose  side  is  s. 

7.  The  perpendiculars  of  two  right-angled  triangles  are 
a  and  b,  the  sum  of  their  bases  c,  and  the  sum  of  their  hypote- 
nuses a  minimum.    What  is  the  base  of  each? 

8.  If  the  solidity  of  a  cylinder  is  2  tt  and  its  surface  a  mini- 
mum, what  is  its  diameter? 

9.  What  is  the  height  of  the  largest  cylinder  which  can  be 
inscribed  in  a  cone  whose  altitude  is  a? 


ON  VARIABLE  QUANTITIES  95 

10.  What  is  the  altitude  of  a  maximum  rectangle  inscribed 
in  a  triangle  whose  base  is  h  and  perpendicular  height  a? 

11.  What  is  the  altitude  of  the  largest  cylinder  that  can  be 
cut  from  a  paraboloid  whose  axis  is  a? 

67.  It  has  been  shown  that  the  value  of  a  single  variable 
which  will  render  its  function  a  maximum  or  a  minimum,  is 
found  by  making  the  first  ratal  coefficient  of  the  function  equal 
to  zero ;  hence  it  is  evident  that  the  value  of  each  variable  of  a 
function  of  two  or  more  variables  is  also  to  be  found  by  mak- 
ing the  first  partial  ratal  coefficient  of  the  function,  relative 
to  that  variable,  equal  to  zero:  that  is,  if  u  =  f  {x,y),  the 
value  of  X  which  will  render  the  function  a  maximum  or  a 
minimum,  is  found  from 

du 

dx 

du 

and  of  3;  from  =  0, 

dy 

whence  all  the  values  of  x  and  3;  can  be  found,  which  will 
render  u  a  maximum  or  minimum. 

It  has  also  been  shown  that  the  second  ratal  coefficient  of  a 
function  of  a  single  variable  is  negative  when  the  function  is  a 
maximum  and  positive  when  it  is  a  minimum ;  for  like  reasons, 
the  values  of  x  and  y,  found  from  the  first  partial  rates  of 
u  =  f  (x,  y),  when  substituted  in  the  second  partial  rates,  must 
give  each  a  negative  value  when  m  is  a  maximum  and  a  posi- 
tive value  when  w  is  a  minimum. 

The  second  partial  rates  of  m  =  /  (x,  y)  are 

d^u    d^u      d~u  d^u 

-,  and 


dx^     dy^     dxdy  dydx 

or,  since  the  last  two  expressions  are  equal  (see  Art.  21),  only 
the  following  need  be  used,  namely 

d^u      d^u  d^u 

,  ,  and  , 

dx^     dy-  dxdy 

each  of  which  must  be  negative  when  m  is  a  maximum  and 
positive  when  m  is  a  minimum. 

The  process  is  similar  when  there  are  three  or  more  inde- 
pendent variables. 


96  AN  ELEMENTARY  TREATISE 

Let  u  =  ax^y-  —  x^y-  —  x^y^, 

the  partial  rates  of  which  are 

du 

=  Zax-y-  —  ^x'^y-  —  Zx-y^ 

dx 

du 

and  =  2ax^y  —  2x^y  —  3x^y^. 

dy 

Making  these  equal  to  zero,  it  will  be  found  that 

1  1 

x  =  —  a  and  -y  =  —  a. 

2  3 

The  second  partial  rates  are 
d^u 


=  6axy^  —  12x^y^  —  6xy^, 
dx^ 

d^u 

=  2ax^  —  2x'^ : —  6:1:^^ 

dy^ 

■       d^u 

and  =  6ax~y  —  8x^y  —  9x^y^. 

dxdy 

Substituting  in  these  the  values  of  x  and  y,  the  results  are 

d^u  a*   d^u  a*  d^u  a* 

—  and 


dx^  9     dy^  8  dxdy  6 

a  a        _  _ 

therefore,  when  ;ir  =  —  and  -y  =  — ,  w  is  a  maximum  and  equal 
2  3 

a« 

to  . 

432 

EXAMPLES 

The  volume  of  a  rectangular  solid  is  s.   What  is  the  length 
of  each  side  when  its  surface  is  a  minimum? 

Let  X,  y,  and represent  the  lengths  of  the  sides,  and  u 

xy 


its  surface. 

2s         2s 

Then 

u  =^  2xy  4- + • 

X          y 

ON  VARIABLE  QUANTITIES  97 

The  first  partial  rates  of  this  are 

du  2s  du  2s 


dx  X-  dy  y"^ 

whence,  by  making  them  equal  to  zero,  it  will  be  found  that 

x^s^'^  and  y  =  s^^^. 

The  second  partial  rates  are 

d-u         As       d^u         4s  d^u 

and 


dx^        x^       dy^        y^  dxdy 

Since  each  of  these  are  positive,  m  is  a  minimum  when  each 
side  is  equal  to  s^^^. 

The  semi-diameter  of  a  sphere  is  r.  What  are  the  lengths 
of  the  sides  of  the  greatest  rectangular  parallelopipedon  that 
can  be  cut  from  it? 


PART  TWO 
THE  INVERSE  METHOD 


PART  TWO 

THE  INVERSE  METHOD 


DEFINITIONS  AND  ILLUSTRATIONS 

68.  In  Part  One  the  function  is  given  to  find  the  rate ; 
herein  the  rate  of  the  function  is  given  to  find  the  function. 

69.  The  method  of  passing  from  the  rate  to  the  function — 
that  is,  the  process  of  restoring  the  function  of  which  the  rate  is 
given — is  called  integration,  and  the  restored  function  is  called 
the  integral  of  that  rate. 

The  integral  of  a  given  ratal  expression  is  indicated  by  the 
character  J  placed  before  it,  as 

f  (2axdx  -f-  hdx), 

showing  that  the  integral  is  required. 

70.  There  can  be  only  one  rate  of  a  given  function,  but 
there  may  be  more  than  one  function  answering  to  a  given 
rate.  This  is  obvious,  since  x'^  and  x'^  -\-  a  have  the  same  rate, 
viz.,  2xdx.  Therefore,  in  integrating,  a  constant  term  must  be 
added  to  the  integral.  This  term  is  usually  represented  by  C ; 
thus  the  integral  of 

du  =  2axdx  is  u^^ax  -\-  C. 

C  is  called  an  arbitrary  constant,  and  the  integral  before 
the  value  of  C  is  known  is  called  an  incomplete  integral.  In 
the  solution  of  a  real  problem,  however,  the  value  of  C  may  be 
determined  from  the  known  conditions  of  the  problem  and 
consequently  a  complete  integral  obtained. 

For  illustration  take  the  ratal  equation  of  the  straight  line, 
dy  =  adx, 
whence  y=^ax-\-C. 

If  the  straight  line  passes  through  the  origin  of  the  coordi- 
nates, then  y  =  0  when  x  =  0;  hence  C  ^  0,  and  the  complete 
or  true  integral  is 

y  =  ax. 


102 


AN  ELEMENTARY  TREATISE 


But  if  the  straight  line  cuts  the  axis  of  ordinates  at  a  dis- 
tance from  the  origin  equal  to  b,  then  for  x  =  0,  y  =  b,  conse- 
quently C  =  b,  and  the  true  integral  is 

y^  ax  -{-  b. 

71.  Of  the  triangle  ABC,  let  AC  be  represented  by  x,  BC 
by  2ax,  CD  by  dx,  and  the  area  of  ABC  by  A ;  then 

dA  =  2axdx, 
whence         A  =  ax~  -\-  C. 

But  when  x^O,  ^  =  0,  conse- 
quently C  =  0;  therefore  the  true 
integral  is         A^^ax~.  (1) 

It  will  be  observed  that  the  same 
is   true   for  the   triangle  AEF  when 
the  area  of  AEF — that  is 


A'  ^ax^. 


(2) 


Now  if  ^^n  in  (1)  and  ;r  =  w  in  (2),  representing  the 
area  of  EBFC  by  A'',  then 


A' 


A  —  A^  =  anr  —  am^. 


This  process  is  termed  integrating  between  limits.  In  the 
present  case,  the  integral  of  2xdx  is  taken  between  the  limits 
of  ^  =  w  and  x^n,  m  being  called  the  inferior  limit  of  x  and 
n  the  superior  limit.  The  sign  of  this  method  is  placed  before 
the  given  rate;  thus    (''Xdx,  X  being  a  function  of  x.     If 

=  0,  then  the  sign  becomes   (^. 

Simple  Algebraic  Rates 
72.   According  to  the  rules  under  Art.  10,  if 


m 


u  = ,  du  =  ax^dx : 

therefore,  it  is  seen  that  the  function  corresponding  to  the  rate 


ax^^dx  is,  by  Art.  70, 


ax" 


n+  1 


+  C: 


ax" 


that  is  Cax'^dx^ +  C. 

•^  n  +  1 

Hence  the  following  rule : 


ON  VARIABLE  QUANTITIES  103 

The  integral  of  a  monomial  rate  is  equal  to  the  constant 
factor  into  the  variable  with  its  exponent  increased  by  unity, 
divided  by  the  exponent  thus  increased,  plus  a  constant  term. 

This  rule  is  applicable  whether  n  is  positive  or  negative,  a 
whole  number  or  a  fraction,  except  when  n  =  — 1,  for  then 

ax''*'^         ax^-'^        ax^         a 


n-\-l       1  —  1         0  0 

But  when  n^  —  1, 

adx 
ax"dx  =  ax'^dx  = , 

X 

which  is  the  rate  of  log  x,  by  Art.  27, 

adx 
therefore  f =  a  log  ;tr  +  C. 

''       X 

Hence  the  integral  of  a  fractional  rate  whose  numerator  is 
the  rate  of  the  denominator  multiplied  by  a  constant,  is  equal 
to  the  constant  into  the  Naperian  logarithm  of  the  denominator, 
plus  a  constant  term. 

Since  a  constant  quantity  retains  the  same  value  through- 
out the  same  investigation,  it  can  be  placed  outside  the  sign 
of  integration,  as  a( x"dx. 

73.  Since  the  rate  of  a  function  composed  of  the  sum  or 
difference  of  any  number  of  terms  containing  the  same  inde- 
pendent variable  is  the  corresponding  sum  or  difference  of 
their  rates  taken  separately  (see  Art.  11),  it  follows  that  the 
integral  of  a  ratal  expression  composed  of  the  sum  or  difference 
of  several  terms  is  equal  to  the  corresponding  sum  or  difference 
of  their  respective  integrals ;  thus 

J  {ax'^dx-{-bdx — nx^~'^dx)=  aJx~dx-\-bJdx —  n  J  x^-^dx= 

1 

-  ax^  -\-  bx  —  x^  +  C. 
3 

From  this  it  is  evident  that  a  polynomial  of  the  form 

du  =  (a  ±  bx  zt  cx^  ±  etc.)"  dx, 

in  which  m  is  a  positive  whole  number,  can  be  integrated  by 
raising  the  quantity  within  the  parenthesis  to  the  nth  power, 
multiplying  through  by  dx,  then  integrating  each  term  sepa- 
rately. 


104  AN  ELEMENTARY  TREATISE 

Let  du^  {a -\- bxY  dx; 

then         du  =  a^dx  -\-  Za^bxdx  -j-  Zah'^x^dx  +  b^x^dx, 
whence    u  ^^J  (a^dx  +  Sa^bxdx  -j-  Sab^x^dx  -\-  b^x^dx)  = 

3  1 

a^x  4-  —  a-bx~  4-  ab~x^  +  —  b^x^  +  C. 
2  4 

When  the  rate  is  of  the  form 

du^  (x^  -\-  ax  -\-  b)^  {2xdx  -\-  adx), 

in  which  n  is  an  integer  or  fraction,  positive  or  negative,  and 
when  the  quantity  within  the  last  parenthesis  is  the  rate  of  that 
within  the  first;  then 

M  =  f  {x^  -\-  ax  -[-  &)"  {2xdx  -\-  adx)  = 

1 

(x""  -\-ax-{-  by^^  +  C. 

n  -\-  I 

This  case  is  substantially  the  same  as  that  of  a  monomial 
rate  (see  Art.  72)  and  is  similarly  inapplicable  under  the  same 
condition:  viz.,  when  the  exponent  w  =  —  1,  for  then 

1  1 

{x^  -\-ax-\-  bY^^= {x^  +  a;ir  +  by-^  = 


n-\-l  1  —  1 

{x^-\-ax^by        1 


^  —  ^  00  ; 

1  —  1  0 

but  when  n  =  — 1, 

{x"^  -\-  ax  -\-  by  {2xdx  +  adx)  = 

2xdx  -f-  odx 


(x-  -y  ax  -\-  b)~^  (2xdx  -j-  adx) 


x^  -\-  ax  -\-  b 

in  which  the  numerator  is  the  rate  of  the  denominator;  there- 
fore 

2xdx  -f-  adx 

u=  C  (x^  -\-  ax  -\-  b)-^  {2xdx  -\-  adx)^  C = 

x^  -\-  ax  -\-h 

log  {x^  ^ax^b)  -\-C. 

74.    To  determine  the  integral  of  a  binomial  rate  of  the 
form 

(/m  =  (a  -|-  bx"")"^  x'^-'^dx : 

that  is,  one  in  which  the  exponent  of  the  variable  without  the 
parenthesis  is  less  by  unity  than  that  of  the  variable  within. 


ON  VARIABLE  QUANTITIES  105 


Assume  a  -\-  bx"  =  y, 

and  taking  the  rate  nbx"~^dx  =  dy 

or  x"    dx  ■■ 


hence  du  =  3;'' 


nb 

dy        y"^dy 
nb  nb 


/iiJH+l 


therefore  by  Art.  72,  u  = 1-  C ; 

nb  (ni  -\-  I) 

or,  substituting  for  3;  its  value, 

u  = 1-  C. 

nb  (m  -\-  1) 

Hence  the  integral  of  a  binomial  rate  in  which  the  ex- 
ponent of  the  variable  without  the  parenthesis  is  one  less  than 
that  within,  is  equal  to  the  binomial  factor  with  its  exponent 
increased  by  unity,  divided  by  the  exponent  thus  increased 
into  the  product  of  the  exponent  and  coefficient  of  the  variable 
within,  with  a  constant  term  added  to  the  result. 

If  the  rate  is 

(a  +  bnx''-'^)  dx 

du  = , 

2  {ax  -\-  bx^^y^ 

1 

or  du=^ —  {ax  -\-  bx")~''^^  («  +  bnx^~^)dx, 

it  will  be  seen  that  the  quantity  within  the  last  parenthesis  is 
the  rate  of  that  within  the  first ;  therefore,  by  Art.  73, 

1 

u^J  —  {ax  -{-  bx"")-^^"  {a  +  bnx'^-^)  dx=  {ax  -\-  bx'^y^ 


If  the  rate 

is 

adx 

du —                , 
b  ±  ex 

by  making 

b  zt  cx  =  y, 

then 

± 

cdx 

±  dy 

=  dy,  or  dx  = 

c 

therefore 

±  ady 
du —                : 

cy 


106  AN  ELEMENTARY  TREATISE 

consequently,  by  Art.  72, 

a 

z*  =  dz  —  log  y  -\-  C, 
c 

or,  substituting  for  y  its  value, 

a 

Jt=  ±  —  log  {h  ±i  ex)  -j-  C. 
c 

EXAMPLES 

ax^dx 

1.  f/w  = 

2 

2.  du  =  (:r-  4"  ^^Y  i^^dx  -\-  adx) 

3.  du=  {1  -{-  ax)~^  2xdx. 
axdx 


4.  du 


{x^  H-  a^) 
5.  du=  {a-\-  bx^y^  mxdx  -\- 


c  -\-  X 

Simple  Circular  Rates 
75.  Referring  to  Art.  31,  it  will  be  seen  that 
M  =j"cos  ;rfl?;r  =  sin  ;r 

dx 

u  ^  r =  tan  X 

cos^  X 
u  =  f —  sin  xdx  =  cos  x 

dx 

u  =  r — =  cot  X 

sin^  X 
u=^  C  sin  xdx  =  vers  x 

tan  xdx 

u  =  J =  sec  X,  etc. 

cos^ir 

Also  in  (3)  of  Art.  31,  it  is  shown  that  the  rate  of 
sin  x'^  =  n  cos  x^'^dx — 
that  is,  Jn  cos  x^'^dx  =  sin  x"';  ( 1 ) 

hence  it  is  clear  that 

J —  n  sin  x^''^dx  =  cos  x":  ( 2 ) 

If  n=  1,  (1)  and  2  become 

w  =  sin  .r  -f-  C"  and  u  =  cos  x  -\-  C. 


ON  VARIABLE  QUANTITIES  107 

EXAMPLES 


1      dx 
1-   /sin 7 


2/- 


X     X- 
2xdx 


cos^  (1  — x^) 


76.  It  is  shown  in  Art.  33,  making  R  =  l  and  omitting  the 
constant  C,  that 

du 


1.  x  =  C — 

^  (I- 


=  sin"^  u 


du 
2.  x^=^ — =  cos"^  w 


du 
Z.  x  =  r =  vers"^  u 

J    {^Zu  —  u-'Y- 

du 
4.  x^  C =  tan~^  u 

•^    1+u^ 

du 

Let  dx  = ,  ( 1 ) 

(a2_w2p 

and  assume  u^av; 

then  du  =  a4v  and  (a-  —  u^Y^=za{\ — v'^Y^. 

Substituting  these  values  in  ( 1 )  gives 

dv 
dx^ : 

dv 
hence  [see  (1)1,  .jr=  C 

dv                        du  u 

or,  since    = and  v==  — 

du  ,       u 

X  ==  r =  sin~^  — . 

(a^  —  M-)'-  a 

du 
Let  dx  = , 

{2au  —  u^)'^^ 

and  assume  u  =  av; 


—  =  sin"^  V 

V2. 


108  AN  ELEMENTARY  TREATISE 

then 

du  adv  dv 

du  =  adv   and 


{2au  —  u^)  '''2        a  ( 2v — v- )  ^^         (  2v — v~ )  '^^ 

du 

therefore  [see  (3)],  jir^  f =  vers"^^'  = 

{Zv  —  z/-) 

du  u 

r __  vers"^  — . 

{2au  —  u~Y^  a 

du 


Let  dx  = 


a'-^-u^ 
and  assume  u^av;  then  du  =  adv 
du  adv  dv 


and 


a^ -\- u~        a-{l-{-v^)       a  (1 -\- V-) 

therefore 

1  dv  1  du  1  u 

X  =  —  r =  —  tan"^  V  =( =  —  tan"^  — , 

a      1  -\-  v^         a  a^  -\-  u^        a  a 

EXAMPLES 

du 
1.  dx^ 


2.  dx 

3.  dx 


(^c  —  u^y- 

du 

{Au  —  2u-y^ 
du 


5  +  M^ 


4.  dx  =  — + 


du 


(l  —  u^y^        (2u  —  u^y 

Integration  by  Series 

77.  Any  expression  of  the  form 

du  =  Xdx, 

in  which  X  is  such  a  function  of  x  that  it  can  be  developed  into 
a  series  of  the  powers  of  x,  may  be  integrated  in  the  follow- 
ing manner.     Supposing  the  development  to  be 

X  =  Ax''  +  Bx^  -f  Cx''  -f  etc., 


ON  VARIABLE  QUANTITIES  109 

then  multiplying  by  dx  and  integrating  each  term  separately 
give 

ABC 

u  =  CXdx  = x"^^  + x^^^  -f x'^^  +  etc. 

•^  a  +  1  ^+1  c  +  1 

This  method  is  often  the  best,  if  not  the  only  course  to 
pursue,  for  when  the  series  are  rapidly  converging,  an  approxi- 
mate value  of  the  integral  may  be  readily  determined. 

dx 
Let  du  = . 

a  -{-  X 

Then,  developing  by  the  binomial  theorem, 

1  I        X        x~       x^ 

—==(o  +  ;r)-^  =  -  —  -  +  -  —  -  + etc.; 

a  -\-  X  a        a-       a-'       a* 

multiplying  by  dx, 

dx  dx        xdx        x~dx         x^dx 

= — + — +  etc., 

a  -\-  X  a  a-  a^  a* 

and  integrating,  the  result  is 

(Xa^  Jv  jC  %  Jv 

u  =  C =(-  — -f — +  etc.)  +C. 

-^   a  +  x  a         2d'        Za^         4a* 

It  has  been  shown  in  Art.  72  that 

dx 

u  =  ( =  log  (a  4-  ;i;)  +  C ; 

a  -\-  X 
therefore 

X         X^  x^  x^ 

u^los  (a4- x)  ^  —  — -f — +  etc. — 

a        2a'         Za^        4a* 

a  result,  when  a=  1,  the  same  as  found  in  Art.  29. 

dx 

Let  du  = =^(\-irX^)-^dx. 

Developing, 

(1  -^  x^)-^  ==\  —  x^  +  x^  —  x""  -^  ^iz. 

Multiplying  by  dx  and  integrating, 

x^      x^      x'' 
u=C (I  -\-  x^)  dx^x  —  —  +  —  —  —  +  etc. 


no  AN  ELEMENTARY  TREATISE 

It  has  been  shown  in  Art.  76  that 

dx 

r =  tan"^  X : 

^  l^x- 

therefore  w  =  J  ( 1  -\-  x-y^  dx  =  tan"^  x  ^= 

X       x^       x^       x'^ 

13        5/ 

When  x  =  0,  the  arc,  and  consequently  C,  equals  0 ;  there- 
fore 

X        x^       x^       x'^ 
w-=tan"^;r^  —  —  —  +  —  —  —  +  etc. 
1         3        5         7 

dx 

Let  du  = =(1 — x^)-'^'dx. 

(l-x^-r^ 

Developing  and  integrating,  the  result  is 

dx 
w  =  C = 

X  x^  3x^  3  •  5x'^ 

(—  + + + +  etc. )  +  C. 

1         2-3         2-4-5  2-4^6-7 

Referring  to  Art.  76,  it  is  found  that 

dx 
C ^=  sin~^  X ; 

therefore  u  =  sin"^  x  = 

X  x^  Zx^  ?>-Sx' 

(—  + + + +  etc. )  +  C. 

1         2-3         2-4-5         2-4-6-7 

Let  du  = .  ( 1 ) 

(x  —  x~y^ 

Assuming  x  =  v^,  then  dx  =  2vdv 

dx                    2dv 
and         = =  2(1 — v')-^''^  dv. 

{x  —  x^y^      (i_z-2)% 

Developing  2  (1  — v^y^,  multiplying  by  dv,  and  integrat- 
ing give 


ON  VARIABLE  QUANTITIES  111 

J2  (l_z;2)-%  dv  = 

V  v^  3f  ^  3  •  5z;^ 

2  (— + + + +  etc.)  +C; 

12-3         2-4-5         2-4-6-7 

but     J"2  (1 — t;^)-^^  a?z/  =  2  sin"^  z^;  therefore,  substituting  for 
V  its  value,  x^'^,  the  result  is 

dx 

u  =  r =  2  sin"^;l^'''^ 

•^    (;ir  — ;ir^)y^ 

2dx 
Putting  (1)  under  the  form  and  assum- 

(2-2;r  —  4x^)'''^ 

ing  2x  =  V,  then 

2dx  dv 


du^^ 


{2-2x  —  4x^y-  (2v  —  v^y^ 

dv 

but  u=C =  vers"^  v ; 

2dx 

therefore        u  =  f =  vers"^  2x. 

^   {2-2x  —  Ax^y 

EXAMPLES 

1.  du={l  +x^ydx 

2.  du=^  (2ax  —  x^y^  dx 

3.  du=  (a  -\-  x)^  dx 

4.  du=  (x^  —  l)-'^^dx 

Binomial  Rates 
78.   If  the  rate  is  of  the  form 

du=(a-\-  hx-'^y  x'^'dx, 
assume  x  =  v-'^,  then  afjr  =  —  v^dv  and  x^  =  z^""* ;   therefore 

c?M  =  — {a-\-hv''yv-'^-^dv, 
in  which  the  exponent  of  v  within  the  parenthesis  is  positive. 
If  the  rate  is  of  the  form 

du  ={ax^  -\-  hx'^y  x'^dx, 
it  can  be  written  thus,  .y  being  less  than  n : 

flfw  =  (a  +  bx'^-'y  x'^^^'dx, 

in  which  only  one  term  within  the  parenthesis  contains  the 
variable. 


112  AN  ELEMENTARY  TREATISE 

Finally,  if  the  rate  is  of  the  form 

du=  {a-\-  bx'^y  x'^dx, 

in  which  ni  and  n  are  fractional,  by  substituting  for  x  another 
variable  having  an  exponent  equal  to  the  least  common 
multiple  of  the  denominators  of  m,  and  n,  a  new  binomial  rate 
can  be  found  in  which  the  exponents  of  the  variable  will  be 
whole  numbers.    Thus,  if  in  the  rate 

du^  {a^  bx"^' ) »"  x^/^dx, 

v^  be  substituted  for  x,  then,  since  dx  =  6v^dv, 

du  =  6  {a  -\-  bv^Y  v^dv. 

Hence  any  binomial  rate  can  be  reduced  to  one  of  the  form 

du^={a^bx''y  x'^dx,  ( 1 ) 

in  which  the  exponents  m  and  n  are  whole  numbers  and  n  is 
positive. 

When  r,  the  exponent  of  the  parenthesis,  is  a  positive  whole 
number,  ( 1 )  can  be  integrated  as  shown  in  Art.  72> ;  also,  when 
m^n  —  1,  as  shown  in  Art.  74. 

Assuming  a  +  bx"  =  v 

in  (1),  then  (a -\-  bx")'' ^^v^',  (2) 

V  —  a 
whence  x'"-  = 


b 

V  —  a 

and  x"^^^  =  ( )  ('»+i>/™ ; 

b 

hence,  by  passing  to  the  rate  and  dividing  by  m  -\-  1, 

1        V  —  a 


Multiplying  this  by  2  gives 

{a -\- bx'^y  x'^dx  = ( )^^^^y^-^v'-dv;        (3) 

bn  b 

1        V  —  a 
hence  Jm  = ( ym^D/n-iyr^y  (4) 

bn  b 

w  +  1    . 
which  can  be  integrated  when is  a  positive  whole  num- 


ON  VARIABLE  QUANTITIES  113 

m  -{-  1 

ber,  or  when  w  +  1  =  w^  (see  Art.  72).  If is  negative, 

n 

see  formula  D,  Art.  80. 

Let  du=  {a  -\-  bx^Y^  x^dx 

fn  -\-  1  1 

in  which  m  =  5,  w  =  2, =  3,  andr  =  — ;  then  by  sub- 

n  2 

stituting  these  values  in  (4),  the  following  is  obtained: 

du  = (  ^       "  y  v'^'dv  = (z/^/2  —  2av^^'  +  a^z/^^)  dv. 

2b  b  2b^ 

Then,  by  integrating  and  reducing, 

1         v^  2av'         a^v 

b-         /  :)  3 

therefore,  since  v^  (a-\-  bx-), 

1      {a  +  bx'Y       2a{a-{-bx^y 

u^—  { — + 

b'  7  5 

a^  (a-^bx^) 

-^ '-}(a^bx^Y^C. 

m>  -\-  1 

If  is  not  a  whole  number,  ( 1 )  may  be  written  thus : 

n 

du  =  [x""  {ax-''  -\-h)Y  x'^dx  =  {ax-""  -\- bY  x'^^^'^'dx. 

By  substituting  in  the  right  hand  member  of  (3)  m -{- nr 
for  m,  —  n  for  n,  a  for  b,  and  b  for  a,  then 

du=  (b  -{-  ax-'')''  x"'^'"'dx  = 

1  v—b 

( ym+nr+l)/-n-l^rdz;^  (5) 

—  an  a 

m  -\-  nr  -{-  1 

which  can  be  integrated  by  Art.  72  when  is  a 

—  n 

positive  whole  number;   if  negative,  by  formula  D  of  Art.  80. 

79.   Referring  to  Art.  9,  it  is  found  that 

d  {vz)  •=  vdz  -\-  zdv, 


114  AN  ELEMENTARY  TREATISE 

whence,  by  integrating, 

V2^  f  vdz  -\-  ^zdv ; 

hence  ijvdz  =  vz  — ^zdv,  ( 1 ) 

in  which  it  is  seen  that  the  integral  of  vdz  depends  upon  that 
of  zdv. 

Resuming  (1)  of  the  last  article, 

c?M  =  (a  +  hx^'Y  x'>'dx,  (2) 

and  assuming  z=  {a  -\-  bx")^, 

in  which  the  exponent  j  may  have  such  a  value  assigned  to  it  as 
may  be  found  most  convenient;  then,  by  passing  to  the  rate, 

dz^bns  {a-{-  bx" ) «-^  x'^-^dx.  (3 ) 

Again,  assuming  vdz  ^  (a  -\-  bx"Y  x^'^dx, 
and  dividing  it  by  (3), 

zi^ 

bns 
and,  passing  to  the  rate, 

r  {  (m  —  n-\-  1)  (a  +  bx'')'-'*^  x'^-''  -\-  1 

bn  (r  —  s -^  I)  (a -{-bx'^y-' x"^}  dx 

du  = = 

bns 

r    a  {m  —  w+l);ir'»-"+    ] 
b  (m  -\-  nr  —  ns  -\-  1 )  x"^ 

{ }  (a  +  bx'')'-^  dx. 

bns 

Now  let  the  value  of  ^  be  such  that 

fii  -\-  7ir  -\-  I 


m  -\-  nr  —  ns  -\-  I  ^0  or  ^^ 
then     dv 


n 
a  {m  —  n-\-l)  (a -{-  bx")  (-»»-iVn  x'>'-''dx 


b  (m  -\-  nr  -\-  1) 

Substituting  the  values  of  v,  z,  dv,  and  c?^  in  ( 1 )  and  inte 
grating, 

f  (a  +  bx'^Y  x'^'dx  ^= 

(a  +  bx'')''^^x»'-"^^  —  a  (m —  n  -\-l)  f  (a  -\-  bx")''x"'-"dx 


b  {m  -\-  nr  -\-  1) 


ON  VARIABLE  QUANTITIES  115 

in  which  the  integral  of  (2)  is  made  to  depend  upon  that  of 

{a  ^  bx'^y  x'^-^'dx. 

In  a  similar  manner  it  will  be  found  that 

^  {a-\-  hx^'Y  x'^'-'^dx 

depends  upon  ^  {a -\- hx'^Y  x'^-^'^dx; 

and  by  continuing  the  process,  the  exponent  of  x  without  the 
parenthesis  can  be  diminished  until  it  is  less  than  n. 

Hence  the  integral  of  a  binomial  rate  may  be  m>ade  to 
depend  upon  the  integral  of  another  rate  of  the  same  form,  but 
in  •which  the  exponent  of  the  variable  without  the  parenthesis 
is  diminished  by  the  exponent  of  the  variable  within. 

If  the  rate  is  of  the  form 

du^  (a-  —  x^) "^^  x^^''dx, 

1 
substituting  a^  for  a,  —  1  for  b,  2  for  n,  and  —  —  for  r  in 

2 
formula  (A)  gives 

1 
u^f(a^  —  x^Y'^x'^dx^  —  —  (a-  —  X'Y''' ^'"'^  + 

m- 

a^  (m —  1) 

— ^ C  (a^  —  x^Y"^- x'^-^dx.  (a) 

m 

In  this  f  (a^  —  x^)-'''' x^'^dx 

depends  on  f  (a^  —  x^) ~^=  x^'^dx, 

and  this,  by  a  similar  process,  will  be  found  to  depend  upon 

^{a'^  —  x^Y'^'^x'^-^dx, 

1 

and  so  on ;  so  that  after  —  m  operations,  since  m  is  an  even 

number,  the  integral  will  depend  upon 

§{d'  —  x^Y'/''dx,  (4) 

X 

which  is,  by  Art.  76,  sin"'^  — . 

a 

If  c/m  =  (  a-  +  ;ir2 )  -%  x'^dx, 

by  substituting  in  formula  {A),  a-   for  a,  1  for  b,  2  for  n,  and 

1 
—  —  for  r,  the  result  will  be 

9 


116  AN  ELEMENTARY  TREATISE 

1 


u 


= J  ( a^  +  ;ir- )  -^^  x'»dx  =  —  ( a^  ^x^ )  ^^  x^-^ 


m 


a^  (m —  1) 

— ^ ({a^-^x^) -%  x'^-^'dx,  ( h ) 

m 

in  which  J  (a~  -\-  ;r^ )"'''=  x'^dx 

depends  on  J  (a-  -f-  •*'^)'^^  x^^'^dx, 

and  by  continuing  the  process,  when  m  is  even,  the  integral 

will  depend  on 

J  {a- -\- x^y^^  dx.  (5) 

Assuming  v  =  x  -\-  (a-  -\-  x^)"^^, 

x-^  (a^^x^y- 

then     dv  =  dx  -[-  (a~-\-x^)  "'/=  xdx  = dx, 

(a'-{-x'y 

dv  dx 

and  = =  {a^ -\- x-)-"^^  dx ; 

V      i^c?  -\-  x-y 

therefore  ^  {a" -\- x~y^- dx  =  \og  {x  ^  (a^  +  jir^)^}  +  C. 

If  Jm  =  ( 2a;tr  — x'^  y^-'X'^dx,  (  6 ) 

assume    z/=  (2a;ir  —  ;r- ) '^^  ;r'"-^  =  (2o;ir2'«-i  —  ^ir^'w)^^; 

a  {2m  —  1 )  x'^'^'-'^dx  —  mx'^'^-^dx 
then  dv  = = 

a  (2m  —  1  );ir"*"^  dx  mx'^dx 

(2ax  —  x'-y  ~   (2ax  —  x^y 

But  the  last  term  is  equal  to  mdu ;  therefore 

a  {2m  —  1 )  x^^~'^dx 
dv  =■ —  mdu 

{2ax  —  x^y 

dv        a  {2m> —  1)  x"^''^dx 
or  du=^  — -\- . 

m  m  { 2ax  —  x-) ^^- 

Hence,  by  integrating  and  substituting  for  v  its  value,  it 
will  be  found  that 

x'^dx  1 

/ =  —  —  {2ax  —  x^-)  y=  x^-^  + 

( 2ax  —  x~y^  m 

a  (2m  —  1 )  x"^~'^dx 

— -§ ,  (0 

m  {2ax  —  x^y 


ON  VARIABLE  QUANTITIES  117 

x^'^dx  .       x"^-'^dx 

in  which    f depends  on    (' , 

^  {2ax  —  x^y^  ^  {2ax  —  x^y- 

and  this  can  be  found  to  depend  on 

x'^-^dx 

^  (2ax  —  x^y' 

and  so  on;  so  that  after  m  operations,  when  m  is  a  positive 
whole  number,  the  integral  will  depend  on 

dx 

^  (2ax  —  x^y' 

which  is,  by  Art.  76, 

vers"^  — . 
a 

In  order  to  obtain  formulas  when  m  is  negative,  multiply 
formula  (A)  by  b  (m  -\-  nr  -{-  I)  ;  then 

Z?  (w  +  wr  -f  1 )  J  (a  +  bx'^)"-  x^dx  = 

(a  +  bx'^Y'"'^  ^m-n+i  —  ^(^^  —  ^_j_l)j(flr4-  bx'')''  x'^-^'dx. 

Transposing  the  terms  containing  the  sign  of  integration 
and  dividing  by  a  {m  —  n  -\-  1)  give 

J  ( a  +  bx"" )  ^  A-™-'*  dx  = 
(a  +  bx'')''^^  ^m-n^i  —  Ij  (^^  _^  nr  ^  1)  f  {a  -{-  bx^^y  x'^dx 

a  {m.  —  w-j-  1) 
in  which  f  (^^  +  bx'^y  x'^-'^dx 

depends  on  ( {a -{- bx^^)^  x^^^dx. 

In  a  similar  manner  it  will  be  found  that 

^  {a -\- bx'^y  x'^dx 

depends  on  /  (^  +  bx'^y  x'^'-^'dx. 

and,  finally,  the  exponent  of  x  without  the  parenthesis  of  the 
last  term  of  {B)  can  be  increased  until  it  is  less  or  greater  than 
n  and  positive. 

Substituting  a^  for  a,  ±  1  for  b,  and  2  for  w  in  (5),  it  will 
be  seen  that 


{B) 


118  AN  ELEMENTARY  TREATISE 

j  {a"  -\- x^^y  x'^-^dx  = 
{a^±  x^y*^  X"'-''  ±.  (w  +  2r+  1)  J(a2  ±  x^y  x'^dx 

a^  (m  —  1 ) 

80.  Again  resuming 

f/w  =  (a  +  bx")''  x'^dx, 
and  assuming  2  =  x^, 

in  which  such  a  value  may  be  assigned  to  the  exponent  j  as 
may  be  desired;  then,  passing  to  the  rate, 

d2  =  sx'-^dx.  (1) 

Assume  vds  =  (a  -\-  bx^ ) '"  x'^dx ; 

then,  dividing  it  by  (1), 

1 

z/  =  —  (a4-  bx'^y  x^^^*'^, 
s 

the  rate  of  which  is 

1 
dv=^—  {m  —  s-\-l)  {a-\-  bx'^y  x'^-^dx  -{- 
s 

bnr 

(a  +  bx'^y-^  x'^-'^^'dx. 

s 

But  (a -\- bx'^y  =  (a -\- bx"")  (a -\- bx'')'-'' ; 

1 
hence  dv  =  —  {a  (m  —  ^+1)  -\-  b  (m  —  s  -\-  nr  -\-  I)  x""} 
s 

•     (a -^  bx'^y-'^  x'^-'dx. 

Now  let  the  value  of  ^  be  such  that 

m  —  s  -\-  nr  -\-  1  ^0  or  s  =  m  -}-  nr  -\-  I ; 

—  anr  (a-\-  bx'')"-^  x'^-'dx 
then  dv  = . 

m  -{-  nr  -\-  I 

Substituting  the  values  v,  2,  dv,  and  dz  in  (1)  of  the  last 
article,  the  result  is 

u  =  ^  {a  -\-  bx^y  x^dx  = 

(a  +  bx'^y  x"^^^  +  anr  f  (a  +  bx'^y-''  x'^dx 

i— ,  (C) 

m  -\-  nr  -\-  \ 


ON  VARIABLE  QUANTITIES  119 

in  which  J  (^  +  hx'^Y  x^dx 

depends  on  j  {a -\- bx'^y-'^  x'^dx, 

and  this,  by  a  Hke  process,  will  be  found  to  depend  on 

J  (a  4-  bx'^y-''  x'^dx, 

and  so  on,  till  r,  the  exponent  of  the  parenthesis  of  the  term 
containing  the  sign  of  integration,  will  be  reduced  to  less 
than  unity  when  positive. 

To  obtain  a  formula  when  r  is  negative,  multiply  (C)  by 
m  -{-  nr  -{-  1 ,  transpose  the  terms  containing  the  sign  of  inte- 
gration, and  divide  by  anr;  then 

u  ^J  (a  -f-  bx'^y-^  x^'dx  = 

—  (o  +  bx'')''  x"^-"^  +  (m  +  wr  +  1)  r  (a  -f  &jr")'"  x'^dx 

.  (D) 

anr 

In  (D)  ^  {a -\- bx'^y-'' x'^dx 

depends  on  ^  {a  -\-  bx'^y  x'^dx, 

and,  by  repeating  the  process,  can  be  made  to  depend  upon  a 
rate  in  which  the  exponent  of  (o  +  bx^)  will  be  positive. 

EXAMPLES 

Determine  the  integrals  of  the  following : 
1,  du={a  —  x'^Y  x^dx 
x^dx 


2.  du  — 

3.  du  =  —(a-^bx^y'>^x^dx 

4.  du=(l  -^xY^'^x-^dx 

Rational  Fractional  Rates 

81.  Every  rational  fractional  rate  can  be  reduced  to  the  form 

{px"^  -\-  qx'^-'^ -\-  rx  -\-  s)  dx  -\- 

A'x^'-^dx  +  B'x^-^'dx  ....  -\-  R'xdx  +  6"'^^ 

Ax''  +  5;ir"-i -\-Rx  ^s 

in  which  the  exponents  of  the  variable  are  all  positive  whole 
numbers,  and  the  greatest  in  the  numerator  of  the  fraction  is 
at  least  one  less  than  in  the  denominator.  Hence,  since  that 
part  of  the  expression  which  is  not  fractional  can  readily  be 


120  AN  ELEMENTARY  TREATISE 

integrated,  it  only  remains  to  integrate  the  fractional  part,  or 

A'x'^-^dx  +  B'x^'-^dx  .  . .  .  +  R'xdx  +  S'dx 

du  = .  (1) 

Ax''  +  Bx'>-^  ....  -^Rx-\-S 

By  resolving  the  denominator  of  this  fraction  into  factors 
of  the  first  degree,  and  assuming  them  to  be 

X  —  a,  X  —  b,  X  —  c,  etc., 

the  equation  may  be  written  under  the  form 

Edx         Fdx  Gdx  Kdx 

du  = + -  + ....  + -,  (2) 

X — a         X — b        X — c  X — k 

in  which  E,  F,  G,  etc.  are  arbitrary  constants  whose  values  can 
be  determined  in  terms  of  a,  b,  c,  etc.  and  A^,  B' ,  C ,  etc.  by 
reducing  (2)  to  a  common  denominator,  and  comparing  the 
coefficients  of  the  like  powers  of  x  in  the  numerator  of  the 
resulting  fraction  with  those  in  the  numerator  of  (1). 

Hence,  when  no  two  or  more  factors  of  the  denominator  of 
(1)  are  alike,  the  integral  of  (2)  is,  by  Art.  72, 

M  =  £  log  {x  —  a)  -\-  F  log  {x  —  b)  -\- 

G\oz{^x  —  o)....^K\og{^x  —  k)^C.  (3) 

When,  however,  two  or  more  factors  are  equal,  as 
a=b=^  c,  (2)  becomes 

Edx         Fdx  Gdx  Kdx 

du  = + +  ■ ....  + -,  (4) 

X — a         X — a        X — a  x — k 

in  which  E,  F,  and  G  have  the  same  denominator ;  consequently 
these  can  be  represented  by  a  single  constant,  as  in 

Hdx  Kdx 

du  = .  . .  .  -|- . 

(x  —  a)  X  —  k 

Here  it  will  be  seen  that  there  are  two  more  equations  to 
satisfy  than  there  are  arbitrary  constants  to  be  determined; 
this  condition,  however,  can  be  obviated  by  writing  the  equa- 
tion thus : 

Edx  Fdx  Gdx 

du  = -\- -|- .  . . 

(x  —  c)^        (x  —  a)-         X  —  a 

which  retains  the  common  denominator  of  (4). 

In  like  manner,  if  there  are  two  or  more  factors,  as 
(x  —  a)'^,  (x — b)'',  the  equation  can  be  written  thus: 


ON  VARIABLE  QUANTITIES  121 

Edx  Fdx  Gdx 

du  = j- .  . .  .  -f- 


i^x  —  ay  {x  —  a)"'-^  {x  —  by 

Hdx  Kdx 

(x  —  &)'-^  X  —  k 

Edx  Fdx  Edx 


The  terms , in  (5),  also 

(x  —  a)^      (x  —  ay  {x  —  by 

Fdx 


etc.     in     this     equation     are     equivalent     to 

{x — b)"'-^  ' 

E  (x  —  a)-^dx,      F{x  —  ay^dx;      and       E  {x—by^dx, 

F  (x  —  b)-"'^^dx,  etc.  can  be  integrated  by  Art.  74;  and  the 
terms  having  denominators  of  the  first  power,  by  logarithms. 

ax^dx  —  c^dx 
If  du  = ,  (6) 

x^  —  c^x 

the  factors  of  the  denominator  are 

X,  X  —  e,  and  x  -\-  c ; 
ax^dx  —  c^dx 


therefore  du  ■ 


X  (x  —  c)  (x  -{-  c) 


Edx           Fdx             Gdx 
Making       du  = + + ,  (7) 

X  X C  X  -]-  c 

and  reducing  it  to  a  common  denominator  give 

Ex^dx  —  Ec^dx  -\-  Fx^dx  -\-  Fcxdx-{-  Gx^dx  —  Gcxdx 
du^ 


Comparing  the  numerator  of  this  with  that  of  (6),  it  will 
be  found  that 

E^F  -}-G  =  a,Fc  —  Gc  =  0,SindEc'  =  c\ 

1  1 

whence  £  =  c,  F  =  — (o  —  c),and  G= — (a  —  c). 

Substituting  these  values  in  (7)  gives 

1  1 

—  (a  —  c)  dx         — (a  —  c)  dx 
edx       2   ^  2 

du  = -f- -\-  - 


X  -\-  c 


122  AN  ELEMENTARY  TREATISE 

and  integrating, 

1 

u=^c\ogx-\-—{a  —  c)  log  {x  —  c)  + 

1 

—  (a  — c)  log  {x-\-c)  ^C 

1 

=  c  log  ;r  +  —  (c  —  c)  log  (^  —  c)  (;r  +  c)  +  C 

1 

=  c  log  X  -\-  —  (a  —  c)  log  {x'^  —  C-)  -\-  C 

=  c  \ogx  -)-  (a —  c)  log  (;r^  —  c^)'/-  -)-  C. 
If  du  = ,  (8) 

{x-\y-{x  —  2) 

then  [see  (4)  and  (5)] 

Edx  Fdx  Gdx 

du  = + + .  (9) 

{x — 1)-  X — 1  X  —  2 

Reducing  to  a  common  denominator, 

E  (x  —  2)  -\-F  {x^  —  3x-\-2)  -\rG  {x'  —  2x-^  1) 
du  = dx. 

ix—ir(x—2) 

Comparing  the  numerator  of  this   with  that  of    (8),   the 
following  are  obtained : 

E  =  S,  F  =  3,  and  G  =  —  3. 

Substituting  these  values  in  (9)  gives 

5dx  3dx  3dx 

du^=^ -|- 


{x — 1)-         X — 1  X  —  2 

3dx  3dx 

5  {x —  \)~-  dx  -\- 


X  —  1  X  —  2 

and  integrating  by  Arts.  74  and  72, 
u  =  —  S  (.r— l)-^  +  31og  (;r— 1)— 31og(^  — 2)  +  C. 

To  verify  the  principle  set  forth  in  (5),  let 

{ax~-\-  hx  -\-  c)  dx 

du  = . 

(  X  —  ry 


ON  VARIABLE  QUANTITIES  123 

Assume  x  —  r  =  v;  then  x=-v  -[-  r,  dx=^dv,  and 
a  {v^  +  2rv  -\-  r"^)  dv  -{-  b  {v  -[-  r)  dv  ^  cdv 


du 


or,  collecting  like  powers  of  v, 

(ar^  -{-  br  -\-  c)  dv  -\-  (2ar  +  b)  vdv  -\-  av^dv 


and,  reducing, 

(ar'^  -[-  br  -\-  c)  dv        ( 2ar  A-  b)  dv         adv 
du  =  - + + 


Substituting  for  v  and  dv  their  values,  x  —  r  and  dx,  then 

{ar~  -\-  hr  4-  c)  dx        (2ar  -\-  b)  dx         adx 

du  =  - ■ — -{-- + ,    (10) 

(x  —  r)^  (x  —  r)-  X  —  r 

in  which  ar^ -\- br -\- c  is  represented  in  (5)  by  E,  2ar -\- b 
by  F,  and  a  by  G. 

Integrating  (10)  by  Arts.  74  and  72  gives 

1 

u  =  —  — {ar~ -\- br -\- c)  {x  —  r)~^  — 

{2ar -\- b)  {x  —  r)~'^ -\- a\og  {x  —  r). 

82.  When  the  denominator  contains  a  single  pair  of  imagi- 
nary factors,  as  x  -\-  r  -\-  s  \/ —  1  and  x  -\-  r  —  s  \/ —  1 
(whose  product  is  x~  -\-  2rx  -]-  r^  -\-  s^) ,  the  fraction  becomes 

A'x''-^dx  +  B'x^-^dx +  S'dx 

du  = ,   (1) 

(Ax^--  +  Bx^-^ +5")  (x""  -^  2rx  ^  r- -\-  s'~) 

which,  assuming  the  factors  of  the  denominator,  other  than 
the  imaginary  pair,  to  be  x  —  o,  x  —  b,  etc.,  may  be  written 
thus : 

Edx  Fdx  Kdx 

du= + r+----  + r-  + 

X  —  a  X  —  0  X  —  k 

Pxdx  4-  Qdx 
^^ .  (2) 

x'^  -\-  2rx  -f-  ^^  +  -^^ 

By  reducing  this  to  a  common  denominator  and  compar- 
ing the  numerator  with  that  of  (1),  the  values  of  E,  F,  etc., 
also  of  P  and  Q,  may  be  determined. 


124  AN  ELEMENTARY  TREATISE 

All  but  the  last  term  of  the  second  member  of  (2)  can  be 
integrated  by  the  methods  in  Art.  81 ;  therefore  it  will  only 
be  necessary  to  integrate  the  last  term,  which  may  be  put  under 
the  form 

(Px^Q)  dx 
dv^— .  (3) 

{x  -\-  r)'^  -\-  s'^ 

Assume  x-\-r^z;  then  since  dx  =  d2,  (3)  becomes 

{Pz  —  Pr+Q)dz        Pzdz          (Pr—Q)dz 
dv  = = — , 

Z~  -\-  S"  z'^  -\-  s-  z^  -\-  s^ 

the  integral  of  which  is  by  Arts.  72  and  76 

1  Pr—Qz 

v  =  —  Plog  (z^-{-s^)— tan-i  — +  C. 

2  .y  .y 

Therefore,  substituting  for  z  its  value  x  -\-  r,  the  integral 
of  (3)  is  found  to  be 

1                                                          Pr—Qx  +  r 
v^^  —  P  log  {x"^  +  2rx  -\-  r"^  -\-  s^') — tan"^ -|"  ^ 


9 


.f 


or 

Pr  —  Q  X  -\-  r 

v  =  P\og  (x^  +  2rx  +  r'  +  s^Y^  — tan-^ +  C. 

s  s 

(2  —  x)  dx                   (2  —  x)  dx 
If     du=^ = :      (4) 

x^  +1  (^+1)  {x^  —  x+l) 

Edx          (Px  -\-  Q)  dx         Edx 
then  du  = -\- = -\- 

X  -[-  I  X- X  -\-  I  X  -\-\ 

Pxdx  Qdx 

~ — rr+  ^     .1-1  '  (=> 

X'  —  X  -^\  x^  —  .ar-j-l 

whence  it  is  found  that  £  =  1,  P  =  —  1,  and  Q  =  1. 

1 
Substituting  z  for  x  —  — ,  ( 5 )  becomes 

1 

{z-^  —  )dz 
dz  2  dz 

du  = — -I- = 

3  3  3 

z^-  z'-}--  ^2  +  - 

2  4  4 


ON  VARIABLE  QUANTITIES  125 

1 

—  dz 

dz  zdz  2 


3  3  3 

z  +  —        ^2  +  —      ^2  +  — 
2  4  4 


the  integral  of  which  is 

3  1  3  V3  2^V3 

w  =  log(^  +  -)— -log(^2+-)-h  — tan-^-y— 

1 

M  =  log(;r  +  1)— --log  {x^  —  x^  1)  + 

V3  (2;r— 1)V3 

tan"' 


3  3 

or 

;ir+l                   V3               (2;r— 1)V3 
u  =  log + tan-^ +  C. 

When  the  denominator  contains  several  sets  of  imaginary 
factors,  respectively  equal  to  each  other,  the  factor 
X  -\-  2rx  -\-r  -\-  s  will  enter  the  denominator  several  times ; 
hence,  for  that  part  of  the  fraction  containing  only  sets  of 
equal  imaginary  factors,  may  be  put  under  the  following  form, 
thus 

{Ex-^F)dx                        {Gx  +  H)dx 
du= + .  •  •  •  + 

(x^  -f  2rx  +  r=^  +  ^-)'»      (x^  +  2rx  +  r'  +  ^')'"-' 

(Px4-Q)  dx 

^       ^^  (6) 


(x^  -\-  2rx  -\-  r~  -\-  S-) 

The  values  of  the  constants  E,  F,  G,  etc.,  may  be  determined 
as  heretofore  explained;  then  the  integral  of  each  term  taken 
separately. 

Since  the  terms  of  the  second  member  of  (6)  are  all  of 
the  same  general  form,  it  will  only  be  necessary  to  integrate 
the  first  term,  which  may  be  placed  under  the  form 

Exdx  4-  Fdx 
dv  = ■ .  (7) 

{x^- ^  2rx -{- r^- ^  s-'Y' 

Assuming  x  =  z  —  r,  this  expression  becomes 

Ezdz — {Er  —  F)  dz 
dv  = = 


126  AN  ELEMENTARY  TREATISE 

Ez  {z^  +  j^)-'«  ds—{Er  —  F)  {z^  +  s'-)-''^  dz, 
and,  by  Art.  74, 

E  (^-  + J2)i-m 
(Ez  (z^  4-3-)  -">  dz  — . 

2(1  — w) 

By  formula  (D)  of  Art.  80,  f—(Er  —  F)  {z^ -\- s^Y'^'dz 
can  be  made  to  depend  upon   J — (Er  —  F)  (z^ -{- s^)~^  dz, 

(Er  —  F)  z 

which  is  — tan'^  — ,  thus  completing  the  integration 

of  (7). 

From  the  preceding,  it  is  evident  that  the  fraction  can  be 
integrated,  even  when  the  denominator  contains  several  dif- 
ferent imaginary  factors ;  providing,  however,  said  factors 
can  be  determined,  and  this  condition  applies  to  all  fractional 
rates. 

EXAMPLES 

adx 
1.  du^:=^ 


2.  du 


3.  du^ 


x~  —  a" 
2axdx  -\-  adx 

x^—\ 
2(1  — x^  dx 


{\^x-y 

Irrational  Fractional  Rates 
83.    Any  irrational  fractional  rate  will  admit  of  integration 
when  it  can  be  changed  to  a  rational  form.    Thus,  let 

{x""^  +  ax'^"'  ^h)dx 
du^==^ , 

X  -\-  cx'^^  -\-  e 

and  assume  x  =  z'^ ;  then 

(z^  -\-  az  -\-  b)  Gz'dz         ( 6z^  +  6az^  -^  6bz^) dz 
du  = = , 

z'^  -\-  cz^  -\-  e  z^  -{-  cz^  +  ^ 

which  is  a  rational  form  and  consequently  can  be  integrated 
by  the  methods  explained  in  Arts.  81  and  82. 

When  the  quantity  under  the  radical  sign  is  a  polynomial, 
the  rate  can  not  in  general  be  changed  to  one  of  a  rational 
form.   If,  however,  the  rate  is  of  the  form 

du  =  {a"  -^hx-\-  c^x-)"^  Xdx,  ( 1 ) 


ON  VARIABLE  QUANTITIES  127 

in  which  X  is  a  rational  function  of  x,  it  can  be  changed  to  a 
rate  which  will  be  rational;  thus,  assuming 

{a^ -^bx-^  c^x'y-  =  2  —  cx,  (2) 

then  a^  -\~  bx  -\-  c^x^  =  s'  —  2czx  -\-  c^x"^ ; 

z^  —  d^ 

whence  x^= .  (jj 

2cz+b 

This  value  of  x  substituted  in  the  second  member  of  (2), 

by  reducing,  gives 

cz~  -{-  bz  -\-  a^c 

(a^ -\- bx  ^  c^x^y^  = .  (4) 

^  ^  2cz+b 

The  rate  of  (3)  is 

2  (cz^-\-bz-\-  a^c)  dz 
dx  =  — .  (5)' 

{2cz^by 

(5)  divided  by  (4)  gives 

2dz 

(a'-^bx-\-c^x^y^^dx  = ;  (6) 

^  2cz-\-b 

2Xdz 

hence     du=  (a^  -\- bx -{-  c'^x'^y'^  Xdx  = ,  (7) 

^  2cz^b 

which  is  a  rational  form ;  for,  since  X  is  a  rational  function  of 
X,  it  must  also  be  a  rational  function  of  s ;  that  is,  if  the  value 
of  X  be  substituted  in  X,  it  will  give  the  value  of  X  in  rational 
terms  of  z. 

1 

If  X  =  — ,  then  substituting  this  value  of  X  in  (7),  since 

X 

z^  —  a^ 

x^ [see  (3)1,  (7)  becomes 

2cz+b         ^    ^J    ^    ^ 

2dz  2cz  -\-  b  2dz 

du={ )  ( ^—)  = . 

2cs  '\-  b  z^  —  d^  z^  —  d^ 

The  integral  of  this  is,  by  Art.  80, 

1         z  —  a 
%  =  —  log ; 

a         z  ^  a 

but  from  (2),  z=^  (a^  -{-  bx  -j-  c^a~y^  -\-  ex, 

1          (a^  4-  bx  +  c^x^y-{-  ex  — a 
therefore  m  =^  —  log -["  ^• 

a         {d^ -\- bx -\- e^x'^y^-\- ex -{- a 


128  AN  ELEMENTARY  TREATISE 

If  Z=  1,  the  integral  of  (7)  will  be 

1 

u  =  —  log  {2c2  -\-  b)  ; 
c 
but  [see  (2)], 

2c2+b=^  2c  {a"  -\-bx-\-  c^x^)"^  +  2c''x  +  b ; 

therefore 

1 

u  =  —  \og{2c  (a^ -\- bx -\- c^x^y^ -\- 2c''x -\- b}  +  C. 
c 

Ub  =  0,  then 

1 

u  =  —  \og2c  {(a^  +  c'^x^y^  4-  ex}  +  C. 
c 

If  X^;!:  then  (7)  becomes 

2  (2^  —  a^)  dz 

du  = , 

{2cz^bY 

which  can  be  integrated  by  Art.  81. 

{a" -\- bx -\- x^y  dx 


If  du 


X 

assume  {a^ -\-bx -\- x^Y^  =  x -\- z; 

then  [see  (3)  and  (4)] 

x=^ (8) 

b  —  2z 

and  {a'^bx^x^y  =  — .  (9) 

b  —  2z 

Taking  the  rate  of  (8)  and  reducing  [see  (5)] 

2  (2^-  —  hz^a^)  dz 
du  =  —  — ^^—- .  (10) 

(b  —  2zy 

Multiplying  (10)  by  (9)  gives 

2  (z^  —  bz -^  a'y  dz 

(a^  J^  bx -\- x^y  dx 

therefore  du 


(b  —  2zy 

2  (z^  —  bz -\- a^y  dz 


(b  —  2zyX 
which  is  rational  in  terms  of  z,  as  previously  explained. 


ON  VARIABLE  QUANTITIES  129 


84.   When  the  rate  is  of  the  form 

Xdx 


{c -]- dx  —  x^y^ 
assume  c  =  ab  and  d^a  —  b;  then 

Xdx 


du=^ ~~. 

{ab-\-  (a—b)  x  —  x^y- 

Now,      since     ab -\-  {a — b)  x  —  x^  =^  {a  —  x)(^b-^x), 

assume  V  [(« —  x){b-^x)]  =  {a  —  x)  z;  (1) 

then,  squaring  both  members, 

{a  —  x)  (b  +  x)  =  {a  —  xy2^ 

or  b-\-x^(a  —  x)  z^, 

0-^^  —  ^ 

whence  x^ ;  (2) 

^2  +  1 

and  therefore, 

az^  —  b          a  -\-  b 
a  —  x  =  a  — ^= .  (3) 

^2  +  1  5^  +  1 

Substituting  this  value  of  a  —  x  in  the  second  member  of 

(1),  the  resuh  is 

(a-{-b)z 
^'[(a-x){b-^x)]=  (4) 

z~  -\-  I 

The  rate  of  (2)  is 

2(a-\-b)zdz 
dx  = .  (.  J ) 

Dividing  this  by  (4)  and  reducing  give 

dx  2dz 


yj[{a  —  x)  {b-^x)-\  z'  +  l 

Therefore,  muhiplying  both  members  by  X,  it  is  found  that 

Xdx  2Xdz 

du  = = , 

^[(a  —  x)  (b-^x)]         ^^  +  1 

which  is  rational  in  terms  of  z,  as  shown  in  Art.  83.    When 
X  =1,  this  becomes 

2dz 


du^ 


z^-\-l 


130  AN  ELEMENTARY  TREATISE 

Hence  u^2  tan"^^:  -|-  C. 

V  [(a  —  x)  (b^x)]  dx 

If  du  = ,  (6) 

Ji. 

then,  proceeding  as  before,  it  will  be  found  that 

du  = ^— I— ^ ,  (7) 

X  (^2  +  1)^ 

which  is  also  a  rational  fraction. 

(x  —  x^)'''^  dx 
Let  du  = .  (8) 

(i  —  xy 

1 

Here  a=l,  b  =  0,   (l  —  xY  = ,   (x  —  x^y^  = 

(z^  +  iy 

-YTV,  and  dx=         \  \        [see  (2),  (3),  (4),  and  (5)]; 

therefore,  substituting  these  values  in  (8)  and  reducing,  it  will 
be  found  that 

2z'^dz  2dz 

du  = ^2  di 


+  1  ^^  +  1 

the  integral  of  which  is 

u^2z  —  2  tan"^^  -|-  C. 

X 

Substituting  for  z  its  value,  ( )'/% 

1  — X 

u  =  2{ )%_2tan-i  ( ^)v.  _|_  C. 

1  — X  1  — X 

EXAMPLES 

1.  du=  (x^  -\-  a)'/^  dx 
dx 


2.  du 

3.  du 


3  (x  —  x~)^^^  dx 


x"" 

Transcendental  Rates 
85.    Simple  rates  of  this  class,  which  admit  of  direct  inte- 
gration, have  been  previously  treated ;  a  few  of  those  whose 
integrals   are   less   readily   obtained   will   now   be    considered, 
omitting  the  constant  C. 


ON  VARIABLE  QUANTITIES  131 

Let  du  =  Xa^dx, 

in  which  X  is  an  algebraic  function  of  x,  and  its  wth  ratal  co- 
efficient is  constant,  represented  by  A  in  the  formula. 

dX  dX' 

Assume  v^^  X,  dz  =  a^^'dx,  2ind— — =^X^,  =^X'',  etc., 

dx  dx 

then  dv  =  dX  and  s  = 


log  a 

These  values  of  v,  z,  dv,  and  a?^;  in  ( 1 ) ,  Art.  79,  give 

Za^  1 

(Xa'^dx^ — j'dXa'', 

log  a         log  a 

1  X'a*-  1 

fdXa-'^  — ~-\- -j'dX'a^, 


log  a  (loga)2  (logo)- 

1  X''a^  1 

and  CdX'a^^ — CdX^'a^,  etc.; 

(loga)^  -^  (loga)^  (loga)«   -^ 

from  which  the  following  is  obtained : 

X  X'  X"  A 

u=^a'={ — + ± }.    (1) 

log  a         (logo)-         (loga)^  (logo)" 

EXAMPLE 

d%^  {hx"^  +  cx^)aF. 

Here  X  =  hx"^  -\-  cx'^,  the  ratal  coefficients  of  which  are 

2bx  +  4cx'',  2x  +  Ucx^,  24cx,  and  24c  =  A. 

Substituting  these  values  in  ( 1 )  gives 

bx^  -\-  ex*        2bx  -j-  4cx^ 

u^^a""  { — -j- 

log  a  (loga)^ 

26  +  Ucx^  24cx  24c 

+  - —}' 


(loga)3  (log  a)*  (loga)  = 

from  which  it  will  be  seen  that  when  the  greatest  exponent  of 
X  is  even,  the  sign  of  the  last  term  of  ( 1 )  will  be  positive,  and 
when  it  is  odd,  the  sign  of  the  last  term  will  be  negative. 

If  du  =  x"^a''dx, 

then  X  =  x"^,  whose  ratal  coefficients  are  {m  being  a  positive 
number)  : 

fyix"^~'^,  m  (m — 1)  x^~^,  m  (m — 1)  (m  —  2)  x"^~^,  etc. 


132  AN  ELEMENTARY  TREATISE 

Substituting  these  values  in  ( 1 )  gives 


mx 


m-i 


u^a^  { — + 

log  a  (loga)- 

m  (m — 1)  x"^~^  m  (m — 1)  (m  —  2)  jr"*"^ 


i^ogay  (logc)^ 

m  (m  —  1)  ....  1 


)•  (2) 


(loga)'«+i 

If  m  be  negative  or  fractional,  then  develop  a^  by  Mac- 
laurin's  theorem,  Art.  24,  multiply  both  members  by  x"^  and 
integrate. 

86.  When  the  rate  is  in  the  form  of  a  logarithm,  as 

du  =  x^  log  xdx, 

assume  v  =  log  x  and  dz  =  x'^dx ; 

dx  .jr"+^ 

then  dv  = and  z 


X  n  -\-\ 

These  values  of  v,  z,  dv,  and  dz'va  ( 1 ) ,  Art.  79,  give 

x^^^  ^«+i       dx 

du  =Cx'"  log  xdx  = log  X  — J  ( ) ; 

n-j-l  n  -\-  \     X 

x"--"^         dx           x'^dx             .ar""-^ 
but  /( ) =  f == ; 

jr"+^                          1 
therefore  u  = ( log  x  — ) . 

n  -\-  1  w  -j-  1 

Let  du^  (logx)''dx,  (1) 

in  which  w  is  a  positive  integer. 

Assume  v=  (log.*")^  and  dz  =  dx; 

dx 
then  dv^=n  (log x)^~'^ and  z  =  x; 

X 

and,  by  substitution, 

du^^J (log x)"  dx  =  x  (log.*:)"  —  wJ(log.^)""^  dx; 
but  — n  f  (log  x)"-'^  dx  = 

—  nx  (log^ir)"^"^  -}-  n  (n —  1)  r(log;r)"~^  dx, 
and  n  (n — I)  f  (log x)^~^  dx  = 


ON  VARIABLE  QUANTITIES  133 

n  (n —  1)  X  (logx)"--^  —  n  (n —  1)  (n  —  2)  J(log;i;)"-^  dx; 

whence  u^^^x  {(log x)^ — n  {log x)^'^ -{- 

n  (n— 1)  {logx)^-^  ....  -\-n  (w— 1)  (....  1)},     (2) 

in  which  the  last  sign  will  be  plus  when  n  is  even,  and  minus 
when  n  is  odd.  If  n  be  negative;  that  is,  if  du  =  (log  x)'^  dx, 
assume  dv^dx  and  z={logx)~^,  and  proceed  as  before; 

dx 

u,  however,  will  be  found  to  depend  on  the  integral  of , 

log^ 

sometimes  called  Soldner's  integral,  which  can  be  obtained  by 
series. 

If  du^^x""- {logxY  dx, 

assume  y  =^  x'"^*'^ ; 

then  log  3/  =  {m  -\-  1 )  log  x 

1 
or  (logJir)»=  ( )»  (log3;)*». 

w  -f-  1 

Therefore,  since  ^3;=  (w  -)-  1)  x^dx, 

1 

or  x^dx  = dy, 

m,  -\-  \ 

1 

d%=^x'^  {logxYdx=  ( )«+i  {'^ogyYdy, 

m  -\-  1 

the  integral  of  which  is  the  same  as  that  of  (1)  multiplied  by 

m  -\-  I 

If  du  =  (log  x)""  Xdx, 

X-^dx                  Xr,dx 
assume  jXdx  =  X-^,  J =  X„,  j'-^ =  X^,  etc. 

X  X 

and  z^^(log:ir)"  and  dz^Xdx; 

dx 
then         dv^:^n  { log x)'^~'^ and  2  =  (Xdx  =  X^. 

These  values  of  v,  z,  dv,  and  dz  substituted  in  (1),  Art.  79, 
give  u^  {logxY  X^  —  w  (log;i;)"-^  Xo  + 

n{n—l)  {logxY'^X  _  ....  w  (w— 1)  (....  1)X      .   (3) 


134  AN  ELEMENTARY  TREATISE 

If  the  integrals  of 

X^dx    X^dx             Xndx 
Xdx,  ,  ,   .... , 

can  be  found  in  finite  terms,  the  proposed  rate  will  have  an 
exact  integral. 

Let  du=  (log  x)^  (1  -\- x^)  dx. 

1 
Here  w  =  3,  J  Xdx  =  x  -\-  —  .r^  =  X^, 

X^dx  1 

/ =  x^-x'  =  X„ 

X  9 

^X,dx  1  ^X.dx  1 

r =  x  -r- .«■■'  =  X„  and   (" ^  x  4- x^  =  X,. 

^     X  27  ^     X  81 

Substituting  these  values  in  (3),  the  result  is 

1  1 

u={\ogxY  (^  +  —  ^3)— 3  (logji;)^  {x^  —  x^)  + 

1  1 

6(log;r)  {x^-—x^-)—6{x^—-x^-). 

Complex  Circular  Rates 

87.    Let  du^  s'm"^  X  cos"  xdx,  (1) 

and  assume  sin  ;ir  ^  7/  ; 

then  cos    x=(l — z/^)'''^  and  cos  xdx=^dv, 

dv  dv 

whence  dx  ■ 


cos  .^r         (1 — v^Y^ 
Substituting  these  values  of  v  and  dv  m  ( 1 ) ,  gives 

du  =  v'''  (l—v^-)^"-^''-dv.  (2) 

Assuming  cos  x^z, 

then  will  du  =  —  ^"^  (1 — s-)^"'-^^^- ds.  (3) 

These  can  be  integrated  by  Art.  78:  (2),  when  m  is  a  posi- 
tive odd  integer  and  m  positive  or  negative,  integral  or  frac- 
tional; (3),  when  m  is  a  positive  odd  integer,  and  n  positive 
or  negative,  integral  or  fractional.  When  these  conditions  do 
not  exist,  they  can  be  integrated  in  many  cases  by  one  of  the 
formulas  A,  B,  C,  or  D. 


ON  VARIABLE  QUANTITIES  135 

In  du  =  sin^  x  cos^  x  dx, 

m  =  2  and  n^^3;  therefore  (2)  becomes 

du  =  v^  (1  —  V')  dv, 

1  1  ^      v^ 

hence  m  =  —  v^  —  —  v^  =  (S  —  3z/-) . 

3  5  15 

or,  substituting  the  value  of  v, 

sin'  X 

u-=  (5  —  3  sin-  x) . 

^  ^      15 

(3)  also  becomes 

du  =  —{l—z^y-z^^dz, 

which  by  formula  A  can  be  made  to  depend  on 

^{\—z^y-zdz, 

which  is  integrable  by  Art.  74. 

In  du  =  sin  x"^  dx, 

assume     sin  ;i;  =  z/ ;      then,     since     cos  x  =  {\  —  v-)"''     and 

dv 
dx^ , 


cos  X 

du^  (1  —  v^)-'^'  V'dv, 

which,  when  w  is  a  whole  number,  either  positive  or  negative, 
by  the  application  of  formula  A  or  B  may  be  made  to  depend 
on  (1 — v^Y^^^  dv  or   (1 — v'^Y^'^vdv. 

The  first  of  these  can  be  integrated  by  Art.  75  and  the 
second  by  Art.  74. 

If  du  =^  tan^  xdx, 

let  tan x^v;  then 

dv 
dx^ 


1  +^2 
V^dv 


and  du - 

l+v- 

a  rational  fraction. 

tan  xdx 
If  du^ 


sm^  X 

sinx 

then,  since  tan  x  = ,  by  substitution  the  result  is 

cos  X 


136  AN  ELEMENTARY  TREATISE 

dx 


du- 


xdx 


sin  X  cos  X 
therefore  u  =  log  tan  x ; 

(see  Art.  Z2). 

Let  du  =  tan"^  xdx. 

Now      d  (x  tan"^  x)  =  tan"^  xdx  -\- 

l-{-x'' 

xdx  1 

and  r =  —  loff(l+^"); 

•^   1  +  ;^2        2 

1 

therefore    .       u^xtain~^x  —  —  log(l+;r^). 

Let  du  =  X  sin"^  xdx, 

and  assume 

Xdx  =  dz  and  sin"^  x  =  v,  also  f  X(/;i:  =  X^ ; 

then  c/z/  ^  (1  —  x^) '^^  dx  and  z  =  fXdx  =  X^. 

Therefore  u  =  X^  sin~^  ;ir  —  ("X^  (1  —  ^r^)"'/^  dx, 
in  which  the  integral  of  the  proposed  rate  is  made  to  depend 
upon  that  of  another,  whose  form  is  algebraic. 

A  similar  process  will  apply  to  any  of  the  following  forms : 

X  cos~^  X,  X  tan"^  x,  X  cot~^  x,  etc., 

since  the  rates  of  cos  "^  x,  tan~^  x,  cot~^  x,  etc.,  all  depend  upon 
the  integral  of  an  algebraic  expression. 

Examples 

1.  du^  a^x'^dx 

x^dx 

2.  du 


log^  X 

3.  du  =  sin^  x  cos^  xdx 

4.  du^  X  cos~^  xdx 

Bernouilli's  Series 
88.    Bernouilli's  series  expresses  the  integral  of  any  rate  of 
the  form 

du  =  Xdx, 

in  which  X  is  a  function  of  x,  in  terms  of  X,  its  ratal  co- 
efficients, and  X. 


ON  VARIABLE  QUANTITIES  127 

To  obtain  this  series,  assume 

X^v  and  dx^==^dz; 

then  dv  =  dX  and  ^  =  ^. 

Substituting  these  values  in  (2)  of  Art.  79,  the  result  is 

j'Xdx  =  xX — jxdX, 

or,  since  dx  is  included  in  dX, 

dX 

i'Xdx  =  xX  —  j  xdx  ( ) . 

•^  -^  dx 

dX 
And  assuming  =  v  and  jra(;ir  ^=  dz ; 

(/^Z  x^ 

then,  since  ofz/  = and  z  = , 

dx"^  1  •  2 

by  substituting  these  values  as  before, 

dX  X'       dX  x^dx      d^X 

—  (xdx  ( )  =  — ( )  +  f ( ) . 

^  dx  1-2      dx       ^   1-2        dx' 

In  a  similar  manner,  it  will  be  found  that 

x^dx     d^X  x^         d'X  ^    x^dx      d^X 

^  1-2      ~d^    ~  1-2-3       dx'-  '^  1  •  2  ■  3      dx' 

x^        dX 

therefore,     by    the     substitution     of     xX  — ( )  + 

•^  1-2        dx 

x'         d'-X 

( ) — etc.,   the   integral   of    (1)    is   found   to   be 

1-2-3   ^dx'  ^  ^  ^   ^ 

x^      dX  x'         d'X 

u  =  xX  — ( )  + ( )  —  etc. 

1-2      dx  1  •  2  •  3       dx- 

This  series  was  obtained  by  John  Bemouilli  in  1694  and  is 
probably  the  first  general  development  discovered ;  it  is,  how- 
ever, but  a  particular  case  of  Taylor's  theorem,  discovered  in 
1715.  Such  expressions  as  log  (1  +  x),  sin  .ir,  and  others  can 
be  readily  developed  into  a  series  by  Bernouilli's  theorem,  as 
shown  by  him. 

Let  du  =  (1  -\-  2x  -{-  3x^ )  dx, 

in  which  (1  +  2  +  3x^)  represents  X ;  then 
xX  =  x  -{-  2x^  -j-  3x', 


138  AN  ELEMENTARY  TREATISE 

X'       dX  x^-         d^X 

— ( )  =  —  X-  —  Zx^,  and ( )  =  x^ : 

1-2      dx  1-2Z      dx' 

therefore 

u  =  X  -\-  2x~  -\-  Zx^  —  X"  —  Zx^  -{-  x^  =  X  -\-  X-  -{-  x^ . 

EXAMPLES 

1.  du={\  -j-x^y^dx 

dx 

2.  du^ 

Successive  Integration 
89.    In  the  expression 

d^u  =  (^^  -j~  ^^^)  dx'^, 

two  integrations  are  required  to  determine  the  primitive  func- 
tion, or  u  in  terms  of  x.    Placing  the  expression  under  the  form 

d'^u 

=  x^dx  -f-  ax^dx, 

dx 

and  integrating, 

dw        X-        ax^ 

—  =  — H-  —  +  C. 

dx         A         ?) 

Multiplying  through  by  dx  and  integrating  again, 

x^  x'^ 

u  = + +  C-,x  4-  C. 

4-5        3-4 

The  foregoing  may  be  written  thus : 

d^u 

dx- 

du 

=  /,  (x)  +  Ci  and  m  =  /g  (x)  -\-  C-^x  -f  C^. 

dx 

From  the  preceding  it  will  be  seen  that,  if 

dnu  =  f  (x)  dx", 

by    taking    n    successive    integration    the    following '  will    be 
obtained, 

w=/   ^  (x)  + 

C,x''-^                            C.x"-- 
1 I   -. U....C     x  +  C  . 

1-2  ....  (w—1)         1-2  ....  (7z  — 2)  "-^ 


ON  VARIABLE  QUANTITIES  139 

The  nth  integral  of 

d^'u^^f  {x)  dx"" 

may  be  represented  thus  : 

u=ff  (x)dx''. 

d^'u 
Developing  ^/  {x) 

dx"" 

by  Maclaurin's  theorem  (Art.  24),  the  result  is 

d^u  df  (x)  X         dH  (x)  x"'        dH  ix)  x^ 

=  A^  +  4-  +  etc. 

dx""  dx  1  •  2dx'^  1  •  2  •  Zdx"^ 

df  {x)      d'-f  {x) 

Now,  by  substituting  for  A,  ,  ,  etc.,  their 

dx  dx^ 

values  as  shown  in  Art.  24,  then  multiplying  by  dx  and  integrat- 
ing n  successive  times,  plus  a  constant  each  integration,  the 
result  will  be  a  series  expressing  the  value  of  u  in  terms  of  x. 

d^u             1 
Let  = , 

dx'^        I  ^  x 

the  development  of  which  is 

d'^u 

^1  —  X  -\-  x~  —  x^  -\-  etc. 

dx^ 

Integrating  this  as  explained,  gives 

x^  x^  x^ 

+  T-— ^— —  etc.  + 


2-3-4        2-3-4-5       3-4-5-6 

C^x^        C.x^ 
-Z-r  +  -T-  +  C,x  +  C, 


EXAMPLE 

d'^u  =  6dx^  -{-  36xdx^  -\-  30x'^dx~. 

Note  :  In  successive  integration,  it  sometimes  becomes  ex- 
pedient to  integrate  between  limits,  especially  when  there  are 
two  or  more  independent  variables.  For  instance,  in  the 
equation  of  the  circle,  y^  =  1  —  x~,  x  can  never  be  greater 
than  1  nor  less  than  zero. 


140  AN  ELEMENTARY  TREATISE 

Integration  of  Partial  Rates 

90.  Partial  rates  are  obtained  with  reference  to  one  variable 
only,  or  with  reference  first  to  one  variable,  then  to  another, 
etc.,  (see  Art.  22). 

Of  the  first  class,  as 

du  =  Zx^ydx, 

the  integral  is  u  =  x^y, 

or,  since  the  primitive  function  may  contain  terms  in  y  alone, 
an  arbitrary  quantity  must  be  added,  as  Y ,  a  function  of  y,  as 
such  terms  will  disappear  in  passing  to  the  rate ;  thus, 

u  =  x^y  -\-Y-\-C. 

This  class  of  partial  rates  can  be  expressed  generally  thus : 

f/"w  =  /  {x,y,  z,  etc.)  dx'^. 

Taking  one  of  the  second  class,  as 

d~u  =  9x^y^dxdy  -\-  2xdxdy, 

and  integrating  first  with  respect  to  x,  then  with  respect  to  y, 

du  =  Zx^y~dy  -\-  x^dy-\-Y  and  u  =  x^y^-\-  x^y-\-J Ydy-\-X-\-  C. 

This  class  of  partial  rates  can  be  expressed  generally  thus : 
d^u^f  {x,  y,  z,  etc.)  dx'^dydz^,  etc., 

in  which  m  is  equal  to  the  sum  of  the  exponents  of  the  rates  of 
the  independent  variables ;  that  is, 

m  =  n-\-r-\-s-\-  etc. 

Let  d^u  ==  6xy"dxdy.  ( 1 ) 

The  integral  of  this  with  respect  to  x  is 

du  =  3x^y^dy  (2) 

or,  since  there  may  have  been  a  term  containing  y  alone  in  (2) 
which  would  disappear  in  (1)  by  passing  to  the  rate, 

du  =  Sx^y^dy  -\-Y. 

Integrating  again  with  respect  to  y,  it  will  be  found  that 

u  =  x~y^-\-fYdy-^X. 

EXAMPLES 

1 .  d^u  =  arx^dy-  +  ydy^ 

2.  d^u  =  x^ydxdy^ 


ON  VARIABLE  QUANTITIES  141 

Integration  of  Total  Rates 
91.    Let     du  =  f^{x,y)  dx-\-f2ix,y)  dy,  (1) 

of  which  the  partial  rates  are 

du=f^  {x,y)  dx 
and  du^f2(x,y)dy. 

Dividing  the  first  by  dx  and  the  second  by  dy  give 
du 

-r^h{^,y)  (2) 

dx 

du 

and  -—  =  f^_(x,y)  (3) 

dy 

Taking  the  rate  of  (2)  with  respect  to  y  and  dividing  by 
dy;  then  the  rate  of  (3)  with  respect  to  x  and  dividing  by  dx, 
the  results  are 

d^u 

=  h(^,y) 

dxdy 

d^u 

and  ———  =  f^(x,y). 

dydx 

Now,  as  is  shown  in  Art.  23,  in  order  that  ( 1 )  be  integrable, 

d~u  d^u 

must  equal  , 

dxdy  dydx 

that  is  /s  (x,y)  =f^  (^,y). 

This  is  termed  the  test  of  integration. 

If  du  =  f  (x,  y,  z)  dx  -\-f  {x,  y,  s)  dy  -\-  f  (x,  y,  s)  dz, 
in  order  that  this  expression  be  integrable  the  following  condi- 
tions must  be  fulfilled :  viz., 

d^u  d^u        d^u  d^u        d~u  d^u 

dxdy        dydx     dxdz         dzdx      dydz         dzdy 

and  similarly  if  there  are  four  or  more  independent  variables. 

Let  du  =  {Zx^y""  ^y^l)dx-\-{2x^y^x^a)dy,  (4) 
the  partial  rates  of  which  are 

du=  {Zx'y"^ -\- y -\- \)  dx  (5) 

and  du  =  {2x^y  -\-  x  -\-  a)  dy;  (6) 

whence  are  obtained  the  following, 

d^u 

=  6x^y  -\-  1 

dxdy 


142  AN  ELEMENTARY  TREATISE 

d~u 

and  =  6x^y  -\-  1, 

dydx 

which  fulfill  the  conditions  stated  above;  therefore  (4)  is 
integrable. 

It  will  be  seen  that  the  original  function  must  have  con- 
tained all  the  terms  in  x  indicated  in  (5),  also  all  the  terms  in 
3;  indicated  in  (6).    Now  the  integral  of  (5)  is 

u-^x^y-  -\-  xy  -{-  X  (7) 

and  of  (6)  u  =  x^y^  -\-  xy  -\-  ay,  (8) 

but  it  will  be  observed  that  the  terms  in  (8)  containing  x  are 
also  included  in  (7),  and  therefore  should  be  omitted  in 
integrating;  consequently  the  integral  of  (4)  is 

u  =  x^y^  -|-  ;r3;  -|-  ;ir  -|-  03;. 

Let  du  =  ay-dx  -\-  2xdy,  (9) 

of  which  the  partial  rates  are 

du  =  ay~dx  and  du  =  2xdy, 

from  which  are  obtained 

d^u  d^u 

=  2ay  and  =  2, 


dxdy  dydx 

which  are  not  equal ;  therefore  (9)  is  not  integrable. 

Let  du  =  (2xy  -{-  2^  -\-  l)dx  -\-  (x-  -\-  3y^z  -f-  ^)  ^3*  + 

(2x2  +  y^  +  4z^  ^  b)  dz.  (10) 

It  is  obvious  here,  as  in  rates  of  two  independent  variables, 
that  the  integral  of  the  coefficient  of  dx  must  have  all  the  terms 
containing  x  in  the  original  function ;  therefore,  in  integrating 
the  coefficient  of  dy,  the  terms  containing  x  must  be  omitted, 
and  in  integrating  the  coefficient  of  ds,  the  terms  containing 
both  X  and  y  must  also  be  omitted. 

Proceeding  thus,  it  is  found  that 

u  =  x-y  -|-  ^^^  -\-  ^  -\-  y^^  +  QJ  +  ■s'*  +  b2. 

EXAMPLES 

du=  {2xy  -\-  3x^n)  dx  -j-  {2xy  +  ^)  dy 

ydx  xdy  xydz 

du  = -j" + 

a  —  z  a  —  z         (a  —  z) 


ON  VARIABLE  QUANTITIES  143 

3  2^1:3;  3xy^  —  x"^ 

du==^—  {x^  —  y-)dx  — dy  + as 

2  z  z^ 

du^  (sin  3;  —  3;sinjir)  dx  -{-  (cos;ir  +  x  cosy)  dy 

Homogeneous  Rates 

92.  A  homogeneous  rate  is  one  in  which  the  sum  of  the 
exponents  of  the  variables  is  the  same  in  each  term ;  this  sum 
is  called  the  degree  of  the  rate,  and  is  here  designated  by  n. 

When  such  a  rate  fulfills  the  conditions  given  in  the  last 
article,  the  integral  can  be  obtained  by  substituting,  for  instance, 
X,  y,  z  for  dx,  dy,  dz,  etc.,  in  their  respective  factors  of  the 
functional  rate,  thus  increasing  by  unity  the  exponent  each  of 
X,  y,  z,  etc.  in  its  said  factor;  then  collecting  like  terms  and 
dividing  hy  n  -\-  \. 

To  prove  this,  let 

du  =  Pdx-\-Qdy^Rdz-\-^i(i.  (1) 

be  a  homogeneous  rate,  in  which  P,  Q,  R,  etc.  are  algebraic 
functions  of  x,  y,  z,  etc.  of  the  nth  degree. 

Now  it  is  evident  that  this  must  have  been  deduced  from  a 
homogeneous  algebraic  function  of  the  form 

u  =  P'x-{-Q'y-{-R'z-\- etc.,  (2) 

of  the  degree  n  -\-  I,  since  taking  the  rate  diminished  by  unity 
the  exponent  of  the  variable  so  treated  in  each  term  of  (1). 

Substituting  xy'  for  y,  xz'  for  z,  etc.  in  (2) ,  each  term  in  the 
value  of  u  will  contain  jr"""^,  consequently 

w  =  A^.a^«+^  (3) 

in  which  iV"  is  a  function  of  y' ,  z' ,  etc.,  but  does  not  contain  x; 
hence,  the  rate  of  (3)  with  respect  to  x,  is 

du 

=(n+l)A/';r^  (4) 

dx 

The  rates  of  xy' ,  xz' ,  etc.  with  respect  to  x,  are  y'dx,  z'dx, 
etc.,  and  these  rates  substituted  in  (1)  and  divided  by  dx,  give 

du 

—-  =  P^Qy'^Rz'^eXc.  (5) 

dx 

du 

but  =  (w  +  1)  Nx"",  (4),  therefore 

dx 

(w  4-  1 )  Nx""  =  P  ^Qy'  ^  Rz'  +  etc. 


144  AN  ELEMENTARY  TREATISE 

or,  by  multiplying  by  x, 

(n  +  1 )  iV^«^i  =  Px-{-  Qxy'  +  Rxz'  +  etc. 
Therefore,  substituting  y  for  xy' ,  z  for  xz' ,  etc.,  and  divid- 
ing by  (w  +  1), 

,,      ,        P^  +  Q3'  +  ^^  +  etc. 

W+  1 

or,  since  Nx^^'^'^  =  u  [see  (3)], 

Px -\- Qy -i- Rz -{-  etc. 
2*  = — .  (6) 

n  -\-  1 

EXAMPLES 

Integrate       du=  (Sx  -\-  mxy)  dx  -\-  {x  -\-  mxy)  dy     and 
{nx'^-'^y  +  y)  J.r  -f  (.r"  +  xy''-^)  dy  -\-  {n  -\-  \)  z"". 

Length  of  Curves 
93.  In  case  of  curves  referred  to  rectangular  coordinates,  it 
has  been  shown  in  Art.  46  that 

dz={dx~  +  dy^y^, 
whence  z  =J  {dx'^  -\-  dy'^Y^, 

which  is  a  general  expression  for  the  length  of  a  curve,  or  the 
length  of  any  arc  thereof,  estimated  from  the  origin  of  the 
coordinates  or  some  special  point.  When  the  radical  is  ex- 
pressed in  terms  of  x  and  dx,  or  y  and  dy,  obtained  from  the 
equation  of  the  curve,  its  integral  may  be  determined. 

In  case  of  polar  curves,  the  rate  of  an  arc  is     [see  Art. 
57,  (1)]: 

dz=  (dr~  -\-  r-dv-y^, 
whence  z  =  ^  {  dr-  -f  r^dv^ )  ^% 

which  is  the  general  expression  for  the  length  of  an  arc  of  a 
curve  referred  to  polar  coordinates,  estimated  from  the  pole  or 
some  special  point.  When  the  radical  is  expressed  in  terms  of 
r  and  dr,  or  v  and  dv,  its  integral  may  be  determined. 

Taking  the   circle  whose  radius  is  unity,  its   sine  x  and 
cosine  (1 — x-)^''-,  then 

X 

t  =  tan  z  = , 

(l—xn'' 
whence 

(I  —  x-y^  dx  +  x'- (I  —  x'-y^^  dx                dx 
dt  =  - ~ ^ = .    (1) 


ON  VARIABLE  QUANTITIES  145 

x^  1 

Now  1  +  ^2^1+ = .  (2) 

1  X'         1  —  x"^ 

Dividing  (1)  by  (2),  the  result  is 

dt                    dx 
= =  dz, 

or  dz^{\-\-f")-^dt, 

the  rate  of  an  arc  of  a  circle  in  terms  of  the  tangent  and  its  rate. 

Developing,  dz=  {\  —  f-  ^  t^  —  t''  ^  etc. )  dt 

f        t"        f 
and  integrating,    z=t  —  —  +  —  —  — -[-etc.,  (3) 

which  needs  no  correction,  since  z^O  when  ^  =  0. 

Now,  by  means  of  the  trigonometrical  formula 

2  tan  a 

tan  2a  = ; 

1  —  tan-  a 

1 

when  tan  a  =  — ,  we  find 
5 

120 

tan  4a  = . 

119 

Also,  by  means  of  the  formula 

tan  A  —  tan  5 

tan  (^  —  5)= , 

1  +  tan  ^  tan  5 

120 

when  tan  A  = and  tan  B  =  tan  45°  =  1,  we  find 

119 

1 
tan  (A  —  B)=- 


239 

1 
Hence,  four  times  the  arc  whose  tangent  is  —  exceeds  the 

1 

arc  of  45°  by  an  arc  whose  tangent  is .  In  a  similar  manner, 

239 

1 

we  shall  find  that  twice  the  arc  whose  tangent  is exceeds 

10 

1  .      1 

the  arc  whose  tangent  is  —  by  an  arc  whose  tangent  is  . 

5  515 


146  AN  ELEMENTARY  TREATISE 

Therefore,  if  ^  =  arc  of  45°,  since  (as  has  been  shown  in 
the  paragraph  immediately  preceding) 

1  1  1 

arc  45°  =  8  tan —  4  tan —  tan , 

10  515  239 

by  applying  these  values  to  (3)  and  multiplying  by  4,  since  arc 
180°  ^TT,  the  following  are  obtained: 

1111  \ 

( — + — +  etc.) 

3(10)3        5(10)^        7(10)^  ^ 

1  1  1 

+ — +  etc.) 

515        3(515)2       5(515)=^        7(515)' 

11  11 

i—  ( — + — -fete.) 

'         239       3(239)3       5(239)^        7(239)^ 

Six  terms  of  the  first  line  and  three  each  of  the  second  and 
third  will  give 

7r  =  3.141592653589793. 
The  transcendental  equation  of  the  cycloid  is  (see  Art.  42) 

ydy 


du^- 


(2ry  —  y^y 


Squaring  this  equation  and  substituting  the  value  of  dx^  in 
the  rate  of  the  arc  give 

y-dy'^ 

d^=(dy^  +  - -)\ 

Iry  —  y 

2r 

or,  reducing,  d2^^=^dy{ Y^. 

2r  —  y 

Putting  this  under  the  form 

ds={2ry-  (2r  —  y)-'^dy, 
and  integrating  by  Art.  74, 

^  =  — 2(2r)^  (2r  — 3;)%  +  C, 
or  s  =  —2yy{2r(2r  —  y)}-\-C. 


ON  VARIABLE  QUANTITIES 


147 


A   F 

C 

F 

g 

If 

then 

and  since 

APD 

— 

AI 

50 


Estimating  the  arc 
from  A,  2^  =  0  for 
y  =  0,  consequently 
0  =  —  4r  +  C  or 
C  =  4r,  hence  ^  = 
4r— 2V{2r(2r— y)}, 
which  represents  the 
B  length  of  an  arc  of 
the  cycloid,  estimated 
from  A  to  any  point, 
as  P. 
y=CD  =  2r 

z  =  APD  =  Ar 

.AP  =  DP=4r~  [4r  — 2v'{2r(2r— y)}], 

DP  =  2^/{2r{2r  —  y)}  (4) 

which  represents  the  length  of  the  arc  estimated  from  D  to  any 
point  P. 

By  similar  triangles, 

CD:DL::DL:DE 

or  DL={CDDEy-; 

but,  since  CP  =  2r  and  y  =  PF  =  CE,  DE  :^2r  —  y,  hence 

DL  =  ^y{2r(2r  —  y)} 
therefore,  comparing  this  with  (4),  it  will  be  found  that 
arc  DP  =  2DL  ; 

that  is,  the  arc  of  the  cycloid,  estimated  from  the  vertex  of  the 
axis  CD,  is  equal  to  twice  the  corresponding  chord  DL  of  the 


Y 


generating  circle. 

The     equation    of    the 
logarithmic  curve  is 

X  =  log  y. 

Passing  to  the  rate  and 
squaring, 

dy^ 


dx' 


r 


Ft  a.  ^1 


Substituting  this  value 
of  dx^  in  the  rate  of  the  arc 
and  reducing  give 


148  AN  ELEMENTARY  TREATISE 

y 

Integrating  by  formula  C  of  Art.  80, 

dy 


^=(1+/)^'^+/ 


3;  (1  ^y-y^ 
Integrating  again  by  Art.  83, 

^=(i+r)^-  — log ~  +  C; 

(1  +r)'^—  (1  — 3') 

or,  multiplying  both  numerator  and  denominator  of  the  fraction 
in  the  second  member  by  ( 1  +  3'" ) '"^^  +  (1  — 3')  and  reducing, 

y 

With  C  the  origin  of  coordinates,  when  x^O,  3;  =  1  and 
2  =  0;  consequently 

0-=V2  — log(l+V2)+C, 

or  C  =  — V2  +  log(l+V2), 

therefore 

1  +  (1+3'')^^ 
^=(l-|-3;2)V3_iog ^ ^_V2  +  log(l  +  V2), 

y 

which  represents  an  arc  of  the  logarithmic  curve  AB,  esti- 
mated toward  B  from  the  point  where  it  cuts  the  axis  of 
coordinates. 

The  equation  of  the  spiral  of  Archimedes  is 

V 


r 


7 . 


^  77 

Taking  the  rate  and  squaring. 


dr^  =  - 


Substituting  this  value  of  dr  in  the  rate  of  the  curve  and 
reducing, 

dv 

d2  =  --(i+v^~)y^. 

First  integrating  by  formula  C  of  Art.  80,  then  by  Art. 
83,  (6), 


ON  VARIABLE  QUANTITIES  149 

1 

Z  =  —  [V  (1  -f  t;2)%__log{(l  J^v^y-  —  V)] 
4'nr 

which  represents  the  length  of  any  arc  of  the  spiral  of  Archi- 
medes, estimated  from  the  pole;  no  correction  is  needed,  since 
^  =  0  when  v==0. 

EXAMPLES 

1.  Determine  the  length  of  an  arc  of  the  common  parabola. 

2.  Determine  the  length  of  an  elliptic  quadrant  in  terms  of 
its  eccentricity,  the  semi-major  axis  being  unity  and  the  semi- 
minor  axis  a. 

3.  Determine  the  length  of  an  arc  of  the  logarithmic  spiral, 
estimated  from  the  point  where  r=  1. 

Area  of  Curves 

94.  The  rate  of  the  area  of  a  curve  referred  to  rectangular 
coordinates  is,  by  Art.  47, 

dA  =  ydx, 

which  can  be  integrated  when  the  second  member  is  given  in 
terms  of  y  and  dy,  or  x  and  dx. 

The  rate  of  the  area  of  a  polar  curve  is,  by  Art.  57, 

1 

dA  =  —  r-dv, 

2 

which  can  be  integrated  when  the  second  member  is  expressed 
in  terms  of  r  and  dr,  or  v  and  dv. 

Multiplying  both  members  of  the  equation  of  the  circle 
by  dx  gives 

ydx=  (R^  —  x^y^dx, 

hence  dA={R~  —  x^- )  ^'^-  dx. 

Integrating,  first  by  formula  C  of  Art.  80,  then  by  Art.  76, 

1  i?2  ^ 

A=  —  xiR'-  —  x-Y-^ sin-i— ,  (a) 

2  2  R 

which  requires  no  correction,  since  A^O  when  x==0. 

1 
Making  x  =  R,  since  the  arc  of  sine  unity  is  —  tt, 

1 
A=  —  i?2 
2 


150  AN  ELEMENTARY  TREATISE 

which  gives  the  area  of  a  quadrant  of  a  circle  whose  radius  is 
R ;  therefore  the  area  of  the  entire  circle  is  R^tt. 

The  equation  of  the  ellipse,  referred  to  its  center  and  axis, 
when  both  members  are  multiplied  by  dx,  is 

b 

ydx  =  —  (a-  —  x-)"^^  dx ; 
a 

b 
hence  dA^—  (a^  —  x^y^  dx. 

a 

Integrating,  first  by  formula  C  of  Art.  80,  then  by  Art.  76, 

A  =  -{-x{d^  —  x-y^  +  ^sm-^-},  (b) 

a      Z  Z  a 

which  requires  no  correction,  since  A=^0  when  x=^0. 

1 

If  jir  =  a,  since  the  arc  of  sine  unity  is  equal  to  — tt,  then 

M  2 

^  \       E^  1 

r G FMGor  A=  —  ab-jT, 

\^^~-^-j;l -^y^      which  represents  the  area  of  a  quarter  of 

\v^^___l__,.„^^'      an  ellipse  whose  semi-major  axis  is  a  and 

n'  semi-minor  axis  is  b;  therefore   the  area  of 

j_.  the  entire  ellipse  is  equal  to  abTr. 

'5"^  Comparing  (a)  with  (b),  it  will  be  seen 

that  the  area  of  a  segment  of  the  ellipse,  as 

CDEF,  is  equal  to  the  area  of  the  corresponding  segment  of 

.  b 

the  circumscribing  circle,  CMNF,  multiplied  by  — ;  hence 

a 

b 
area  DEE'D'  =  —  ( area  MNN'M' ) . 
a 

Taking  the  general  equation  of  the  parabola 

yn  =  ax  or  y  =  a^^^x'^^", 
and  multiplying  both  members  by  dx,  the  result  is 

ydx  ==  a^/"x'^^"dx ; 

hence  dA  =  a^^'^x^/^'dx. 

n 
Integrating,    A  = Qi/n^cn+D/n  _|_  c. 

M  -)-  1 


ON  VARIABLE  QUANTITIES  151 

Estimating  the  area  from  the  vertex  of  the  parabola,  A=0 
when  x=^0,  and  consequently  C  =  0;  therefore 

n  n 


n-{-  1  n  -\-  I 

or,  substituting  3;  for  a^^"x'^^"', 

n 

A= -^3^,  (1) 

n  -\-  I 

which  represents  the  area  of  a  segment  of  any  parabola,  and  is 
equal  to  the  rectangle  described  by  the  abscissa  and  ordinate, 

n 
multiplied  by  the  constant  term . 

n  -\-  \ 

li  n  =  2,  (1)  becomes 

2 
A^  —  xy: 
3 

that  is,  the  area  of  a  segment  of  the  common  parabola  is  equal 
to  two-thirds  of  the  area  of  the  rectangle  described  by  the 
abscissa  and  ordinate. 

If  w=l,  (1)  becomes 

1 

A=jxy; 

that  is,  the  area  of  a  triangle  is  equal  to  half  the  product  of  its 
base  and  perpendicular. 

Multiplying  both  members  of  the  equation  of  the  hyperbola, 
referred  to  its  center  and  axes,  by  dx  gives 

b  b 

ydx^—  {x"-  —  a^y^  dx  or  dA=^— {x"-  —  a~y^dx. 
a  a 

Integrating  first  by  formula  C  of  Art.  80,  then  by  Art.  83, 

bx  (x^  —  a^)^^         ab 

A^ ^ ^_ log{^+  (x^---a'y^}^C. 

2a  2 

When  A^O,  x  =  a;  consequently 

ab  log  a 


C 


2 
therefore 

bx  (x^  —  a^y^'         ab  x -[-  (x^  —  a^y^ 

A=-^ ^— — log( ^ '—}, 

2a  2  a 


152 


or,  since 


AN  ELEMENTARY  TREATISE 
b 


a 


{x'  —  d'Y 


y, 


1 


A=^  —  xy  ■ 

2 


ah  bx  +  ay 

log( ^-^). 

2  ab 


Squaring  both  members  of  the  equation  of  the  spiral  of 

^^  .       .  1  . 

Archimedes   (r  =  — )   and  muUiplying  by  —  dv  give 
27r  2 

1  v-dv 
—  r~dv  = , 

2  Stt^ 


or 


dA=^ 


v^dv 


whence,  integrating. 


24  TT^ 


+  c:. 


Estimating  the  area  from  the  pole,  y^  =  0  when  ^'  =  0,  and 
consequently  C  =  0;  therefore 

i7;3 


If  v^2iT,  then 


^=- 


24- 


ON  VARIABLE  QUANTITIES  152 

which  represents  the  area  of  PBA,  described  by  one  revolution 
of  the  radius  vector :  that  is,  the  area  of  the  first  spire  is  equal 
to  one-third  of  the  area  of  a  circle,  whose  radius  is  equal  to  the 
radius  vector  of  the  spiral  at  the  end  of  the  first  revolution. 

If  v  =  Att,  then 

8 

3 

which  represents  the  area  described  by  the  radius  vector  in 
two  revolutions;  but  it  will  be  seen  that  the  radius  vector 
describes  the  portion  PBA  a  second  time ;  therefore,  to  obtain 
the  area  of  PB'A',  the  area  described  by  the  first  revolution 
must  be  deducted :  that  is, 

8  1  7 

area  PBA  =  —  tt  —  —  ■7r  =  —  tt. 
3  3  3 

EXAMPLES 

1.  Determine  the  area  of  the  cycloid. 

2.  Determine  the  area  of  a  segment  of  the  logarithmic  curve, 
lying  between  the  curve  and  axis  of  ordinates,  estimated  from 
the  point  where  the  curve  cuts  the  axis  of  ordinates. 

Surface  Areas  of  Revolution 
95.  For  a  curve  referred  to  rectangular  coordinates,  revolv- 
ing about  the  axis  of  X,  the  rate  of  the  surface  area  of  rotation 
is  (see  Art.  48) 

dS^^Zir  ydz. 

In  case  the  curve  is  revolved  about  the  axis  of  Y,  it  is  evi- 
dent that  the  rate  of  the  surface  area  will  then  be 
dS  =  2Tr  xdz. 

When  the  second  member  of  either  of  these  equations  is 
expressed  in  terms  of  x  and  dx  or  y  and  dy,  the  integral  thereof 
may  be  determined. 

From  the  equation  of  the  common  parabola  it  will  be 
found  that 

ydy 

dx  = . 

p 

Substituting  this  value  of  dx  in  the  rate  of  the  area  of  the 
surface  of  revolution, 

y'^dy- 
dS  =  27ry{- +  df'y\ 


154  AN  ELEMENTARY  TREATISE 

or  dS  =  -^^(y-^p^y^dy. 

P 

Integrating  by  Art.  78, 

3p 
Estimating  the  arc  from  the  origin  of  coordinates,  6'  =  0 

2p-7r 

when  7  =  0:  hence  C=:  — and 

3 

^=-7^{(f'  +  p'r'—p'}, 

3p 

which  represents  the  surface  area  of  revolution  of  the  common 
parabola  for  any  ordinate  3^. 

The  equation  of  the  ellipse  is 

a-y^  =  a^b^  —  b^x^, 

b  a^  —  b^ 

ydz^ — (a-  — x'Y^  dx 

a  a~ 

or,  representing  the  eccentricity  of  the  ellipse  by  e,  by  substi- 
tuting a^e'^  for  a-  —  b^,  since  a^  —  b~  =  d-e^, 

b 

ydz  =  — (a-  —  e-x-y^dx; 
a 

2be  TT     a^ 

therefore  dS  = ( —  —  x'^Y^  dx. 

a         e- 

Integrating,  first  by  formula  C  of  Art.  80,  then  by  Art.  76, 
gives 

be  IT     a^  ab-jT  ex 

6"  = (—  —  x-y^  X  ^ sin-^ , 

a        e^  e  a 

which  needs  no  correction,  since  S ^0  when  x  =  0;  hence 
the  expression  represents  the  surface  area  of  that  part  of  an 
ellipsoid  estimated  from  the  vertex  of  the  minor  axis  and  cor- 
responding to  the  abscissa  x,  the  arc  being  revolved  about  the 
major  axis.     By  making  x^a  and  reducing, 

abiT 

S  ==  b~iT  -{- sin"^  e, 

e 

which  gives  one-half  the  area  of  the  surface  of  the  ellipsoid; 


ON  VARIABLE  QUANTITIES  155 

therefore  if  S'  represents  the  area  of  the  entire  surface,  then 

S'  =  Zb^TT  + sin-i  e. 

e 

When  a^b,  ^  =  0,  and  the  equation  becomes 

S'  =  Zb^TT  +  2&-7r  =  4^  V, 

the  area  of  the  surface  of  a  sphere  whose  semi-diameter  is  b. 

If  the  elHpse  be  revolved  about  its  minor  axis,  then  will 

a 

xds  =  —  {b*  -\-  a-e-y^ ) ^^^  dy ; 

2a  TT 

hence  dS  ^^ (  ^*  +  d^^'y^ )  ^^^  dy. 

b~ 

Integrating,  first  by  formula  C  of  Art.  80,  then  by  Art.  83, 
gives 

S  =  -^  {¥  +  a^e^-y^y^y^ 
b'~ 

^^\og{{b'  +  a'e'~y^y^  +  aey)  +  C. 
e 

Estimating  the  surface  from  the  vertex  of  the  major  axis, 
5  =  0  when  3'  =  0,  in  which  case 

C  =  — log  b- ; 

e 

a-TT 

therefore  5*  = ( ^*  +  a^e-y"^  Y^  y  -\- 

b^ 

7  9  L2 

^^log  {(^^  +  a^e'y'^y  -f  0^3;}  —^^  log  b\ 
e  e 

air  b'Tz  (b"^ -\- a^e^y-Y^  4- aey 

or  5-  =  -(&*  +  aVy)V3  3,  + log{^ ^ '-}. 

b-  e  b- 

Now,  since  b^a{\  —  e^Y^  and   (&- +  a^^^)"''^  =  a,  when 
y^^b,  this  becomes 

feV  ab(l-\-eY' 

S  =  a^ir  + log 

e        ^ab(l  —  eY' 

b^TT         (l-\-eY' 
or  5^  =  a^TT  -\- log 


e  (1—^)% 


156 


AN  ELEMENTARY  TREATISE 


which  represents  half  the  area  of  the  surface  of  a  spheroid. 
If  S'  represents  the  entire  surface,  then 


If  0  = 


■  2aV  -|-  2&^7r{ 
b,  then 


log  (14-^)y^_log(l  — ^)y^ 


}• 


:2&-7r{l    + 


log(l  +  0'''^  — log(l  — ^)' 


Now,  when  a=b,  ^  =  0 ;  but  by  Art.  35, 
log(l  +  ^)%  — log  {l  —  ey- 


1; 


therefore  the  surface  of  a  sphere  whose  semi-diameter  is  b,  is 

S'  =  Ab-7r. 

From  the  equation   of   the   logarithmic    curve,   it   will   be 
found  that 

ydz  ^  ( 1  -f-  y'^y^dy,  hence  dS  =  2  7r  (1  -}-  y^y^  dy. 

Integrating  by  Arts.  80  and  83, 

log[(l+3'^)^/^  +  3']}+C. 

Estimating  the  surface  from  P, 
the  point  where  the  axis  of  ordi- 
nates  cuts  the  curve,  S^O  when 
y^l;  consequently 

C  =  vr{— V2  — log(V2  +  l)}; 
therefore 

s  =  ^{y(i+y'y-h 
log  [{l^f-y  +  y]}— 

-{V2  +  log(V2+l)}.. 
This  represents  the  area  of  the  surface  of  revolution  of  any 
arc  of  the  logarithmic  curve,  estimated  from  P,  as  PC,  and  cor- 
responding to  the  ordinate  y  =  DC,  the  curve  being  revolved 
about  the  axis  of  abscissas  AB. 


EXAMPLES 


1.  Determine  the  area  of  the  convex  surface  of  a  right  conoid 
whose  perpendicular  height  is  a  and  diameter  of  base  is  b. 

2.  Determine   the   area   of   surface    of    revolution    of   the 
cycloid  when  revolved  about  its  base. 


ON  VARIABLE  QUANTITIES  157 

3.  Determine  the  area  of  the  convex  surface  of  a  cubical 
paraboloid,  when  the  axis  of  ordinates  is  the  axis  of  revolution. 
The  equation  is  3'^  =  ax. 

Volume  of  Revolution 

96.   The  rate  of  the  volume  of  revolution  generated  by  an 

arc  of  a  curve  revolved  about  its  axis  of  abscissas  is,  by  Art.  49, 

dV  =  7ry^dx.  (1) 

When  the  second  member  of  this  equation  is  expressed  in 

terms  of  either  x  and  dx,  or  y  and  dy,  its  integral  can  be 

determined. 

When  the  axis  of  Y  is  the  axis  of  revolution,  the  rate  of  the 
volume  of  revolution  thus  generated  is 

dV  =^  IT  x^dy. 

From  the  general  equation  of  the  parabola,  3;"  =  ax,  it  will 
be  found  that 

n 

dx^  —  y'^-^dy. 

a 

Substituting  this  value  oi  dx  in  (1), 

n 
dV  =  —  7r  y^^dy, 
a 

and  integrating,  the  result  is 

n  n         y" 

a  (n  -\-  2)  n  -\-2      a 

or  since  — ^x,     V^ny^  ( —  x)  +  C, 

a  n  -\-2 

which  needs  no  correction,  since  v^O  when  x  =  Q.   If  m=  1, 

1 

then  V  ^  —  TT  y^x, 

which  represents  the  volume  of  a  right  cone  whose  altitude  is  x 
and  y  one-half  of  the  diameter  of  its  base. 

1 

If  w  =  2,  then  V  =  —  Try'-x, 

which  represents  the  volume  of  the  common  parabola. 


158  AN  ELEMENTARY  TREATISE 

The  equation  of  the  elHpse,  when  the  origin  is  at  the  vertex 
of  its  minor  axis,  is 

a- 

x-  =  —{2by  —  y-); 
b^ 
hence  [see  (2)], 

dV  =  —^  (2by  —  /)  dy. 
b" 

aV  1 

Integrating,   V  = {by'^  —  —  3;^)  -\-  C, 

b~  3 

in  which  C  =  0  when  y  =  0;  therefore 

a-TT  1 

V  =  --{bf-  —  -y^). 
b-  3 

li  y=^b,  then 

2 

V  ^  —  a-biT, 

3 

which  represents  the  volume  of  one-half  of  a  spheroid;  hence 
the  entire  volume  is 

4  2 

V'  =  —  a'bTr  =  —  (Za^bir). 
3  3 

But  2a~bTr  represents  the  volume  of  a  cylinder,  whose  altitude 
is  2b  and  the  radius  of  whose  base  is  a;  therefore  the  volume 
of  a  spheroid  is  equal  to  two-thirds  of  the  volume  of  a  circum- 
scribed cylinder. 

The  equation  of  the  hyperbola,  when  the  origin  of  the 
coordinates  is  at  the  vertex  of  the  transverse  axis,  is 

b^ 

3;^  =  —  {x^  -\-  2ax) . 
d~ 

Substituting  this  value  of  3)-  in  (1), 

dV  = {x^  -f  2ax)  dx. 

a- 

feV      1 

Integrating,  V  = ( — x^  -)-  ax-)  -\-  C. 

a^       3 

Estimating  the  volume  of  revolution  from  the  origin  of 
coordinates,  we  have  V  =  0  when  jr  =  0,  and  consequently 
C  ^  0 ;  therefore 


ON  VARIABLE  QUANTITIES  159 

b'-TT      1 

a~       3 
which  represents  the  volume  of  revolution  of  the  hyperbola 
for  any  abscissa. 

The  ratal  equation  of  the  cycloid  is 

ydy 

dx^ . 

{2ry  —  3'")'''^ 

Substituting  this  value  of  af;r  in  ( 1 ) , 

TTj^dy 


dV 


{2ry  —  y'^Y^ 
the  integral  of  which  is 

1  5  y 

V  =^  tt{ —  {2y^  4~  5ry  -\-  15r')  {2ry  —  y^)''^  -j-  —  r"  vers"^ — }. 
6  2  r 

If3;  =  2f,  then       V^  —  r^  vers"^  2, 


or,  since  vers"^  2  ^  tt, 


5 

9 


which  represents  one-half  the  volume  of  revolution  generated 
by  the  cycloid  revolved  about  its  base.     The  entire  volume  is 

F'  =  5rV^ 

EXAMPLES 

1.  Determine  the  volume  of  rotation  of  the  ellipse  when  the 
origin  of  the  coordinates  is  at  the  vertex  of  the  major  axis. 

2.  Determine  the  volume  of  revolution  of  the  logarithmic 
curve  when  revolved  about  its  axis  of  abscissas. 

3.  Determine  the  volume  of  revolution  about  its  axis  of 
abscissas  of  the  curve  whose  equation  is 

y  =  x  (x  -\-  a). 

97.     Let  BDEF  be  a  plane  moving  from  A  toward  X  along 
the  axis  of  X  and  at  right-angles  thereto,  and  let  AC  be  repre- 


160 


AN  ELEMENTARY  TREATISE 


sented  hy  x,  BC  by  y,  and  FC 
by  v;  then  the  rate  of  the 
volume  of  the  solid  thus  gen- 
erated will  be 

dV  =  f  {v,  y)  dx, 
or,      since      y^f{x)       and 

dV  =  f{x)dx        (1) 

This  formula  is  applicable 
to   the  volume   of   any   solid, 
when   the  area   of   the   plane 
BDEF  can  be  expressed  in  terms  of  x  and  dx. 

Determine  the  volume  of  a  right  pyramid  whose  base  is  a 
rectangle. 

Let  the  perpendicular  Aa  be  represented  by  x,  the  side  BC 
by  y,  and  the  side  C\D  by  v ;  also  let  y  =  ax  and  v  =  hy.  Then 
the  area  of  BCDE  will  be 
vy  =  ahx^ ;  hence 

dV  =  abx^dx, 
and  integrating, 

1 

F  =  —  abx^. 

3 

But  abx^  is  the  area  of  BCDE; 
therefore  the  volume  of  the  pyra- 
mid is  equal  to  the  area  of  its 
base  multiplied  by  one-third  of 
its  perpendicular  height. 

Required  the  volume  of  a  parabolic  paraboloid,  the  fixed 
parabola  being  the  semi-cubical  and  the  generatrix  the  common 

parabola. 

Let  x  =  AC,  y  =  BC,  and 
v=^CE  =  CD;  then  for  ABX, 

3;3/2  :=  ax, 

and  for  BCE,  v~  =  by. 

From    these    equations    it 
will  be  found  that  the  area  of 

4 
BDE  is  —  ab'/'x, 
3 


Fi 


g- 


ON  VARIABLE  QUANTITIES 


161 


also 


and  integrating, 


dV  =  —  ab^''^xdx, 
3 

2 
3 


Required  the  volume  of  an  elliptical  ellipsoid,  the  equations 
being  for  the  fixed  ellipse  a^y^  =  a^b^  —  b^x^  and  for  the  gen- 
eratrix a^v^  =  a^c^  —  c^x'^  and  the  origin  of  coordinates  being 
at  the  center. 

Let  A'C  =  a,BC  =  b,  CF  =  c, 
CC'=x,B'C'=y,  and  C'F'=v; 
then  from  the  equations, 

y  =  -{a'  —  x^y% 


(a2_;ir2)v^, 


and     vy 


be 


(^a'—^x^). 


therefore 


Fin.  JS 


be 


But    7rz;3;  =  area    of    B'D'E'F', 


Trbc 


dV  = (a^  —  x'')  dx={7rbe~ x^)  dx, 

IT  be 
and  integrating,         V=^iTbex  — ;r^ 

which  requires  no  correction  since  F  =  0  when  x=^Q;  hence, 
making  ;ir  =  a, 

1  2 

V^Trabe  —  —  irabe^  —  Trabe. 
3  3 

Since  V  is  one-half  the  volume  of  the  ellipsoid,  F',  the  volume 

4 
of  the  entire  solid,  is  —  tt  abe. 

3 

EXAMPLES 

1.  Determine  the  volume  of  an  elliptical  conoid  whose 
altitude  is  a'  and  the  radius  of  whose  base  is  b\ 

2.  Determine  the  volume  of  a  groin  formed  by  the  inter- 
section of  two  equal  semi-cylinders  at  right-angles  to  each 
other,  the  equation  being  y  =  2rx  —  x. 


162 


AN  ELEMENTARY  TREATISE 


Curved  Surfaces  Referred  to  Three 
Rectangular  Coordinates 

98.  To  obtain  a  formula  for  the  volume  of  a  solid  bounded 
by  a  curved  surface  and  referred  to  three  rectangular  coordi- 
nates, X,  y,  and  s,  of  which  z^f  {x,y). 

Let  the  plane  C'CPD  be  para- 
llel to  the  plane  A'ZX  and  the 
plane  EPBB'  parallel  to  the  plane 
YZA'  ■  also  let  A'B' =  C'P'=  x, 
A'C'  =  B'P'  =  y  and  P'P  =  z. 
Represent  dx  by  P'a'  =  c'h'  and 
dy  by  P'c'  =  a'b' ;  then  zdxdy 
will  represent  the  rate  of  the 
volume  of  the  solid;  that  is, 

dW  =  zdxdy.  {A) 

To  obtain  a  formula  for  the 
surface  area,  let  Pa^cb  repre- 
sent dx,  Pc  =  ah  represent  dy, 
and  Nc  =  Ma  represent  dz ;  also 
let  PM  be  a  tangent  to  the  curve 

CPD  Sit  P  (Fig.  60),  PA^  a  tangent 

to  the  curve  BPE  at  P,  and  PQ  a 

perpendicular  to  NM.     Then   will 

PN=  (dx^-  +  dz^y^ 
PM=  (dy^  -\-dz^y^ 
and      NM  =  (dx^ -\-  dy^ )  ^/^ 

From  these  three  equations  the 
following  is  obtained : 

(dx^dy^4-dx'-dz^4-dy^dz-y' 

PQ=- ^-^= ^— ^^ —; 

(dx^^dy^y^ 

but  PQ  ■  NM  =  area  of  NPML  = 

(dx^dy""  +  dx^dz^-  +  dy^dz-y^ ; 

therefore  d'^S  = 

(dx^dy^ -^  dx'-dz^- ^  dy''dz^-y\  (B) 

Required  the  volume,  also  the  surface  area  of  a  sphere,  the 
equation  being 

^2  =  r- —  (x- -{- y-) .      _  (1) 

For  the  volume  [see  formula  (A)],  it  will  be  seen  that 
d^V=  {r^  —  x'^  —  3;^ ) '/^  dxdy  =  zdxdy. 


ON  VARIABLE  QUANTITIES  162 

The  integral  of  this  with  respect  to  y,  between  the  limits 
y^^O  and  y^  (r^  —  x-)^''',  is 

1  3'  1 

2  ^  (,■■'  — x-'y^        4 

y     1 

since  {r-  —  x^)'^^=^y  and  sin  ^  —  =  —  tt. 

3'        2 

Integrating  this  expression  with  respect  to  x  gives 

1  1 

F  =  —  TT  (r^x  —  —  x^)  4-  C, 
4  3 

or  between  the  limits  x  =  0  and  x^r, 

1  1  1 

V  =  —  TT  (r^  —  —  r^)  =  —  r^  TT, 
4  3  6 

which  represents  one-eighth  of  the  volume  of  a  sphere;  there- 
fore the  volume  of  the  entire  sphere  is 

4 

V'  =  —  7rr\ 
3 

Resuming  (1), 

^2  ^  ,^2  ^2  y2^ 

and  taking  the  rate,  first  with  respect  to  x,  then  with  respect 
to  y,  the  results  are 

xdx                             ydy 
dz  =  — and  ds^  — : 


z 


y^dy^                      x^dx^ 
hence  dz~  = and  dz^  = 


z^  z^ 

Substituting  these  values  of  dz^  in  formula  B,  so  that 
dx^dz^        dy^dz^ 


x^dx'^dy'^         y^dy^dx^ 
shall  read  -|- 


z^  z~ 

and  reducing  (since  dz  will  be  eliminated),  the  result  is 

dxdy 
d-S  = —  (x^  -^y^-\-  z^y^ 


164 


AN  ELEMENTARY  TREATISE 


or,  since 

{x-  ^y^-  ^s^-y-  =  r, 

and 

z^  (r-  —  X-  —  y'^y^, 

rdxdy 

J9.C                                       -^ 

{r^  —  x-  —  y^y 

The  integral  of  this  with  respect  to  y,  between  the  Hmits 
of  y  =  0  and  3'^  (r  —  x),  is 

ydx 


dS  ^r  sin~^ 

y={r-  —  x-y% 

1 

therefore,  since  sin~^  1  =  — tt, 


(^r-  —  x-y^ 

y 


1; 


1 


dS  =  —  rrr  dx, 

2 

the  integral  of  which,  between  the  limits  of  ;r  :^  0  and  jr  =  r, 

1 

is  S=^  —  r^TT, 


which  represents  one-eighth  of  the  surface  area  of  a  sphere, 
therefore  the  entire  area  is 

99.  A  body  T,  with  a  uniform  velocity,  proceeds  from  C , 
toward  A,  along  the  straight  line  CA, 
B^  and  a  body  P,  with  a  velocity  which 
is  to  that  of  T  as  1  to  n,  proceeds 
from  B  in  pursuit  of  T  and  always  in 
the  direction  of  T.  Required  the 
equation  of  the  curve  APB,  called  the 
curve  of  pursuit,  which  is  described 
by  P. 

Let  A  be  the  origin,  BC  =  a, 
AS^x,  PS^=y,  and  the  arc  AP=z; 
then  AT  =  nz  and  (see  Art.  40)  the 


subtangent  ST  ^ 


ydx 


dy 


ON  VARIABLE  QUANTITIES  165 

Now,  since 

ydx 
AT  =  AS — ST  =  x 


dy 


ydx 

X  — =  nz. 

dy 


Taking  the  rate  of  this,  regarding  dy  as  constant,  and  re- 
ducing give 

— ^=  ndz ;  ( 1 ) 

dy 

dx^ 

but  dz={dx' ^dy-y^  =  dy  { +  l)'''^ 

dy" 

From  this  and  ( 1 ) ,  the  following  is  found : 

ndy         dx^  d^x 


y  dy~  dy 

dx 

Integrating  by  Art.  84,  regarding  as  the  variable,  the 

dy 

result  is 

—  nlog3'  =  log  {-—+(-— +l)'/^}+C. 
dy  dy- 

dx  dx 

Since  =  tan  SPT,  when =  0,  y=^a]  therefore 

dy  dy 

—  n  log  a^C, 

hence,  transposing  the  value  of  C,  it  is  found  that 

dx  dx^ 

n\oga  —  w  log 3^  =  log  (^  +  (tT  +  1)'''')' 

dy  dy- 

a"  dx  dx- 

or  log-=r.log{— -+  (-— -f  1)''^}; 

3;"  dy  dy- 

a"        dx  dx^ 

hence  —  = +  ( +  1 )  '^'■ 

y"-       dy  dy~ 

By  resolving  this,  it  will  be  found  that 

1  1 

dx^=^  —  a'^y'dy  —  —  a-"y''dy,  (  2 ) 


166  AN  ELEMENTARY  TREATISE 

the  integral  of  which  is 

a"  1 

X  = 3;^-"  — 3;^+",  (3 ) 

2(1  — n)  2a»  (1  +w) 

which    needs    no    correction,    since   3'  =  0    when   x^O,    and 
therefore  is  the  required  equation. 

Dividing  (3)  by  n,  then,  when  y  =  a  and  x  =  AC, 
X  a  a  a 

n         2w  (1  —  n)  2n  {I -\- n)  1  —  n^ 

AC 
or,  since  x  =  AC  and =^APB, 


APB  = .  (4) 

19 
—  n- 

1 

When  n  =  — ,  (3)  and  (4)  become 

-y%  1 

^  =  — -(a  — —  3;), 

4 
and  APB=  —  a. 

3 

From  (1),  (2),  and  (3)  it  will  be  found  that 

1  (fjr  a"  1 

—  {x  —  y )  = 3'''"  + 3''""  ; 

n  dy  2(1— w)  2a"(l  +  >t) 

and  s  = 3'"'+ 3''""'  (5) 

2  (1— w)  2a"  (1  +w) 

1 
which  represents  the  length  of  any  arc,  as  AP.    When  w  =  — , 


(5)  becomes 


1  3f^^ 

^  =  av^yv^  +  -—  //^  =  -— ■  (3a  +  y  ) . 
3a'^  3a^^ 


Date  Due 

i 

1 

y 


BOSTON  COLLEGE 


3  9031   01548848  9 


160491 


ci 


BOSTON  COLLEGE  LIBRARY 

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